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THE  THEORY  AND  PRACTICE 


OF 

SURVEYING. 

DESIGNED  FOR  THE  USE  OF 


SURVEYORS  AND  ENGINEERS  GENERALLY, 


BUT  ESPECIALLY  FOR  THE  USE  OF 

Students  in  Engineering, 


BY 

J.  B.  JOHNSON,  C.E., 

Professor  of  Civil  Engineering  in  Washington  University,  St.  Louis,  Mo. 
Formerly  Civil  Engineer  on  the  U.  S.  Lake  and  Mississippi  River 
’Purveys ; Member  of  the  American  Society  of 
Civil  Etigineers. 


FOURTH  EDITION. 


NEW  YORK : 

JOHN  WILEY  & SONS, 

15  Astor  Place. 

1888. 


Copyright,  1880, 

By  J.  B.  JOHNSON. 


JLsti- 


PREFACE  TO  THE  FIRST  EDITION. 


No  apology  is  necessary  for  the  appearance  of  a new  book 
on  Surveying.  The  needs  of  surveyors  have  long  been  far  be- 
yond the  accessible  literature  on  this  subject,  to  say  nothing  of 
that  which  has  heretofore  been  formulated  in  text-books.  The 
author’s  object  has  been  to  supply  this  want  so  far  as  he  was 
able  to  do  it. 

The  subject  of  surveying,  both  in  the  books  and  in  the 
schools,  has  been  too  largely  confined  to  Land  Surveying.  The 
engineering  graduates  of  our  technical  schools  are  probably 
called  upon  to  do  more  in  any  one  of  the  departments  of 
Railroad,  City,  Topographical,  Hydrographical,  Mining,  or 
Geodetic  Surveying  than  in  that  of  Land  Surveying.  Some 
of  these  subjects,  as  for  example  City,  Geodetic,  and  Hy- 
drographical Surveying,  have  not  been  formulated  hitherto, 
in  any  adequate  sense,  in  either  English  or  any  other 
languasre,  to  the  author’s  knowledge.  In  the  case  of  Geodetic 
Surveying  there  has  been  a wide  hiatus  between  the  matter 
given  in  text-books  and  the  treatment  of  the  subject  in  works 
on  Geodesy  and  in  special  reports  of  geodetic  operations.  The 
latter  were  too  technical,  prolix,  and  difficult  to  give  to  stu- 
dents, while  the  former  were  entirely  inadequate  to  any  rea- 
sonable preparation  for  this  kind  of  work  on  even  a small 
scale.  The  subjects  of  City  and  Hydrographical  Surveying  as 
here  presented  are  absolutely  new. 

Part  I.  treats  of  the  adjustment,  use,  and  care  of  all  kinds 


IV 


PREFACE. 


of  instruments  used  in  surveying,  eitlier  in  field  or  office.*  In 
describing  the  adjustments  of  instruments  the  object  has  been 
to  present  to  the  mind  of  the  reader  the  geometrical  relations 
from  which  a rule  or  method  of  adjustment  would  naturally 
follow.  The  author  has  no  sympathy  with  descriptions  of  ad- 
justments as  mechanical  processes  simply  to  be  committed  to 
memory,  any  more  than  he  has  with  that  method  of  teaching 
geometry  wherein  the  student  is  allowed  to  memorize  the 
demonstration. 

Many  surveying  instruments  not  usually  described  in  books 
on  surveying  are  fully  treated,  such  as  planimeters,  panto- 
graphs, barometers,  protractors,  etc.  The  several  sets  of  prob- 
lems given  to  be  worked  out  by  the  aid  of  the  corresponding 
instruments  are  designed  to  teach  the  capacity  and  limitations 
of  such  instruments,  as  well  as  the  more  important  sources  of 
error  in  their  use.  This  work  is  such  as  can  be  performed 
about  the  college  campus,  or  in  the  near  vicinity,  and  is  sup- 
posed to  be  assigned  for  afternoon  or  Saturday  practice  while 
the  subject  is  under  consideration  by  the  class.  More  ex- 
tended surveys  require  a special  field-season  for  their  success- 
ful prosecution. t 

The  methods  of  the  differential  and  integral  calculus  have 
been  sparingly  used,  as  in  the  derivation  of  the  barometric  for- 
mula for  elevations,  and  of  the  L M Z formulae  in  Appendix 
D.  Such  demonstrations  may  have  to  be  postponed  to  a later 
period  of  the  course.  % 


* Certain  special  appliances,  as  for  example  heliotropes,  filar  micrometers, 
current-meters,  etc.,  are  treated  in  the  subsequent  chapters. 

f At  Washington  University  all  the  engineering  Sophomores  go  into  the 
field  for  four  weeks  at  the  end  of  the  college  year,  and  make  a general  land 
and  topographical  survey,  such  as  shown  in  Plate  II.  At  the  end  of  the  Junior 
year  the  civil-engineering  students  go  again  for  four  weeks,  making  then  a 
geodetic  and  railroad  survey.  Some  distant  region  is  selected  where  the 
ground,  boarding  facilities,  etc.,  are  suitable. 


PREFACE. 


V 


Part  II.  includes  descriptions  of  the  theory  and  practice  of 
Surveying  Methods  in  the  several  departments  of  Land,  Topo- 
graphical, Railroad,  Hydrographical,  Mining,  City,  and  Geo- 
detic Surveying;  Surveys  for  the  Measurement  of  Volumes; 
and  the  Projection  of  Maps,  Map  Lettering,  and  Topographi- 
cal Signs.  The  author  has  tried  to  treat  these  subjects  in  a 
concise,  scientific,  and  practical  way,  giving  only  the  latest  and 
most  approved  methods,  and  omitting  all  problems  whose 
only  claim  for  attention  is  that  of  geometrical  interest. 

In  treating  the  trite  subject  of  Land  Surveying  many  prob- 
lems which  are  more  curious  than  useful  have  been  omitted, 
and  several  new  features  introduced.  The  subjects  of  com- 
puting areas  from  the  rectangular  co-ordinates,  and  the  supply^ 
ing  of  missing  data,  are  made  problems  in  analytical  geometry, 
as  they  should  be.  A logarithmic  Traverse  Table  for  every 
minute  of  arc  from  zero  to  90°,  arranged  for  all  azimuths  from 
zero  to  360°,  to  be  used  in  connection  with  a four-place  loga- 
rithmic table,  serves  to  compute  the  co-ordinates  of  lines  when 
the  transit  is  the  instrument  used.  A traverse  table  com- 
puted for  every  15  minutes  of  arc  is  no  longer  of  much  value. 
The  isogonic  declination-curves  shown  on  Plate  I.  will  be  found 
to  embody  all  the  accessible  data  up  to  1885,  reduced 

from  the  U.  S.  Coast  Survey  chart.  Appendix  A will  be 
found  of  great  value  as  outlining  the  Judicial  Functions  of  the 
Surveyor  by  the  best  possible  authority. 

The  chapter  on  Mining  Surveying  was  written  by  Mr.  C, 
A.  Russell,  C.E.,  U.  S.  Deputy  Mineral  Surveyor  of  Boulder, 
Colorado.  He  has  had  an  extended  experience  in  Hydro- 
graphic  Surveying,  in  addition  to  many  years’  practice  in  sur- 
veying mines  and  mining  claims. 

The  chapter  on  City  Surveying  was  written  by  Mr.  Wm. 
Bouton,  C.E.,  City  Surveyor  of  St.  Louis,  Mo.  Mr.  Bouton 
has  done  a large  proportion  of  the  city  surveying  in  St.  Louis 


vi 


PREFACE. 


for  the  last  twenty  years,  and  has  gained  an  enviable  reputa- 
tion as  a reliable,  scientific,  and  expert  surveyor. 

It  is  believed  that  the  ripe  experience  of  these  gentlemen 
which  has  been  embodied  in  their  respective  chapters  will  ma- 
terially enhance  the  value  of  the  book. 

The  author  also  desires  to  acknowledge  his  indebtedness  to 
his  friend  H.  S.  Pritchett,  Professor  of  Astronomy  in  Wash- 
ington University,  for  valuable  assistance  in  the  preparation 
of  the  matter  on  Time  in  Chapter  XIV. 

Although  the  theorems  and  the  notation  of  the  method  of 
least  squares  are  not  used  in  this  work,  yet  two  problems  are 
solved  by  what  is  called  the  method  of  the  arithmetic  mean 
(which,  when  properly  defined,  is  the  same  as  the  method  of 
least  squares),  which  will  serve  as  a good  introduction  to  the 
study  of  the  method  of  least  squares.  These  problems  are  the 
Rating  of  a Current-meter,  in  Chapter  X.,  and  the  Adjustment 
of  a Quadrilateral,  in  Chapter  XIV.  The  author  has  found 
that  such  solutions  as  these  serve  to  make  clear  to  the  mind 
of  the  student  exactly  what  is  accomplished  by  the  least- 
square  methods  of  adjusting  observations. 

The  chapter  on  Measurement  of  Volumes  is  not  intended 
to  be  an  exhaustive  treatment  of  the  subject  of  earthwork,  but 
certain  fundamental  theorems  and  relations  are  established 
which  will  enable  the  student  to  treat  rationally  all  ordinary 
problems.  The  particular  relation  between  the  Henck  pris- 
moid  and  the  warped-surface  prismoid  is  an  important  one, 
but  one  which  the  author  had  nowhere  found. 

An  earthwork  table  (Table  XL)  has  also  been  prepared 
which  gives  volumes  directly,  without  correction,  for  the 
warped-surface  prismoid.  The  author  has  no  knowledge  that 
sucli  a table  has  ever  been  prepared  before. 

A former  work  by  the  author  on  Topographical  Surveying 
oy  the  Transit  and  Stadia  is  substantially  included  in  this 
oook. 


PREFACE, 


Vll 


The  methods  recommended  for  measuring  base-lines  with 
steel-tapes  are  new ; but  they  have  been  thoroughly  tested, 
and  are  likely  to  work  a material  change  in  geodetic  methods. 

The  author  wishes  to  acknowledge  his  obligations  to  many 
instrument-manufacturers  for  the  privilege  they  have  very 
kindly  accorded  to  him  of  having  electrotype  copies  made  from 
the  original  plates,  for  many  of  the  cuts  of  instruments  given 
throughout  the  book ; persons  familiar  with  the  valuable  cata- 
logues published  by  these  firms  will  recognize  the  makers 
among  the  following:  W.  & L.  E.  Gurley,  Troy,  N.  Y. ; Buff 
& Berger,  Boston,  Mass.;  Fauth  & Co.,  Washington,  D.  C. ; 
Queen  & Co.  and  Young  & Sons,  Philadelphia,  Pa.;  Keuffel 
& Esser,  New  York  ; and  A.  S.  Aloe  and  Blattner  & Adam  of 
St.  Louis,  Mo.  Also  to  Mr.  W.  H.  Searles  for  the  privilege 
of  using  copies  of  plates  from  his  Field-book  for  Tables  L, 
VI.,  and  VII. 

Hoping  this  work  will  assist  in  lifting  the  business  of  sur- 
veying to  a higher  professional  plane,  as  well  as  to  enlarge  its 
boundaries,  the  author  submits  it  to  surveyors  and  engineers 
generally,  but  especially  to  the  instructors  and  students  in  our 
polytechnic  schools,  for  such  crucial  tests  as  the  class-room 
only  can  give. 

J,  B.  J. 

St.  Louis,  Sept.  23,  1886. 


^ Sc* 


TABLE  OF  CONTENTS. 


PAGE 

Introduction j 

BOOK  I. 

SURVEYING  INSTRUMENTS. 

CHAPTER  I. 

INSTRUMENTS  FOR  MEASURING  DISTANCES. 

The  Chain  ; 

1.  The  Engineer’s  Chain 5 

2.  Gunter’s  Chain 5 

3.  Testing  the  Chain 6 

4.  The  Use  of  the  Chain 8 

The  Steel  Tape  : 

5.  Varieties 9 

6.  The  Use  of  Steel  Tapes 10 

Exercises  with  the  Chain  : 

7-17.  Practical  Problems ii,  12 

CHAPTER  II. 

INSTRUMENTS  FOR  DETERMINING  DIRECTIONS. 

The  Compass  : 

18.  The  Surveyor’s  Compass  described  13 

19.  The  General  Principle  of  Reversion. 15 

20.  To  make  the  Plate  perpendicular  to  the  Axis  of  the  Socket 16 

21.  To  make  the  Plane  of  the  Bubbles  perpendicular  to  the  Axis  of  the 

Socket 16 

22.  To  adjust  the  Pivot  to  the  Centre  of  the  Graduated  Circle 16 


X 


cox  TRACTS. 


23.  To  straighten  the  Needle 

24.  To  make  the  Plane  of  the  Sights  normal  to  the  Plane  of  the  Hubbles. 

25.  To  make  the  Diameter  through  the  Zero-graduations  lie  in  the  Plane 

of  the  Sights 

26.  To  remagnetize  the  Needle 

27.  The  Construction  and  Use  of  Verniers 

The  Declination  of  the  Needle: 

28.  The  Declination  defined 

29.  The  Daily  Variation 

30.  The  Secular  Variation. . 

31.  Isogonic  Lines 

32.  Other  Variations  of  the  Declination 

33.  To  find  the  Declination  of  the  Needle 

Use  of  the  Needle  Compass  : 

34.  The  Use  of  the  Compass 

35.  To  set  off  the  Declination 

36.  Local  Attractions 

37.  To  establish  a Line  of  a Given  Bearing 

38.  To  find  the  True  Bearing  of  a Line.- 

39.  To  retrace  an  Old  Line 

The  Prismatic  Compass  : 

40.  The  Prismatic  Compass  described 

Exercises  : 

41-44.  Exercises  for  the  Needle  Compass 38, 

The  Solar  Compass  : 

45.  The  Burt  Solar  Compass 

46.  Adjustment  of  the  Bubbles 

47.  Adjustment  of  the  Lines  of  Collimation 

48.  Adjustment  of  the  Declination  Vernier 

49.  Adjustment  of  the  Vernier  on  the  Latitude  Arc 

50.  Adjustment  of  Terrestrial  Line  of  Sight  to  the  Plane  of  the  Polar 

Axis 

Use  of  the  Solar  C»,'  mpass  : 

51.  Conditions  requiring  its  Use 

52.  To  find  the  Declination  of  the  Sun 

53.  To  correct  the  Declination  for  Refraction 

54.  Errors  in  Azimuth  due  to  Errors  in  the  Declination  and  Latitude 

Angles 

55.  Solar  Attachments 

Exercises  with  the  Solar  Compass  : 

56-59.  Practical  Problems 53, 


■AGR 

17 

17 

17 

18 

18 

20 

20 

21 

23 

29 

29 

34 

36 

36 

37 

37 

37 

38 

39 

39 

41 

41 

42 

43 

43 

44 

44 

45 

49 

52 

54 


CONTENTS.  xi 


CHAPTER  III. 

INSTRUMENTS  FOR  DETERMINING  HORIZONTAL  LINES. 

PAGE 

Plumb-line  and  Bubble  : 

60.  Their  Universal  Use  in  Surveying  and  Astronomical  Work 55 

61.  The  Accurate  Measurement  of  small  Vertical  Angles 58 

62.  The  Angular  Value  of  one  Division  of  the  Bubble 58 

63.  General  Considerations 59 

The  Engineer’s  Level  : 

64.  The  Level  described 60 

65.  Adjustment  of  Line  of  Sight  and  Bubble  Axis  to  Parallel  Positions.  63 

66.  Lateral  Adjustment  of  Bubble  67 

67.  The  Wye  Adjustment 67 

68.  Relative  Importance  of  Adjustments ^ 68 

69.  Focussing  and  Parallax 68 

70.  The  Levelling-rod 70 

71.  The  Use  of  the  Level 71 

Differential  Levelling  : 

72.  Differential  Levelling  defined 72 

73.  Length  of  Sights 73 

74.  Bench-marks 74 

75.  The  Record 75 

76.  The  Field  work 76 

Profile  Levelling  : 

77.  Profile  Levelling  defined 77 

78.  The  Record 78 

Levelling  for  Fixing  a Grade  : 

79.  The  Work  described 81 

The  Hand  Level: 

80.  Locke’s  Hand  Level 81 

Exercises  with  the  Level  : 

81-85.  Practical  Problems 82 

CHAPTER  IV. 

INSTRUMENTS  FOR  MEASURING  ANGLES. 

THE  TRANSIT. 

86.  The  Engineer’s  Transit  described 83 

87.  The  Adjustments  slated 86 

88.  Adjustment  of  Plate  Bubbles 86 

89.  Adjustment  of  Line  of  Collimation 87 


xn 


CONTENTS. 


I'AGK 

90.  Adjustment  of  the  Horizontal  Axis 87 

91.  Adjustment  of  the  Telescope  Hubble 89 

92.  Adjustment  of  Vernier  on  Vertical  Circle 89 

93.  Relative  Importance  of  Adjustments 89 

Instrumental  Conditions  affecting  Accurate  Measuremen  is  : 

94.  Eccentricity  of  Centres  and  Verniers 90 

95.  Inclination  of  Vertical  Axis  91 

96.  Inclination  of  Horizontal  Axis 92 

97.  Error  in  Collimation  Adjustment 93 

The  Use  of  the  Transit  : 

98.  To  measure  a Horizontal  Angle 93 

99.  To  measure  a Vertical  Angle 94 

100.  To  run  out  a Straight  Line . . 95 

101.  Traversing 97 

The  Solar  Attachment  : 

102.  Various  Forms  described 99 

103.  Adjustments  of  the  Saegmuller  Attachment 102 

The  Gradienter  Attachment  : 

104.  The  Gradienter  described 104 

The  Care  of  the  Transit  : 

105.  The  Care  of  the  Transit 104 

Exercises  with  the  Transit  : 

106-114.  Practical  Problems 105-107 

the  sextant. 

115.  The  Sextant  described 108 

116.  The  Theory  of  the  Sextant no 

117.  The  Adjustment  of  the  Index  Glass in 

118.  The  Adjustment  of  the  Horizon  Glass in 

119.  The  Adjustment  of  the  Telescope  to  the  Plane  of  the  Sextant in 

120.  The  Use  of  the  Sextant 112 

Exercises  with  the  Sextant  : 

121.  121^.  Practical  Problems 112,  113 

The  Goniograph  : 

122.  The  Double-reflecting  Goniograph 113 

CHAPTER  V. 

THE  PLANE  TABLE. 

123.  The  Plane  Table  described 117 

124.  Adjustment  of  the  Plate  Bubbles 119 

125.  Adjustment  of  Horizontal  Axis 119 


CON  TEN  7' S. 


XIU 


PAGE 

126.  Adjustment  of  Vernier  and  Bubble  to  Telescopic  Line  of  Sight.. . 119 
The  Use  of  the  Plane  Table: 

127.  General  Description  of  its  Use 120 

128.  Location  by  Resection 123 

129.  Resection  on  Three  Known  Points 123 

130.  Resection  on  Two  Known  Points 124 

131.  The  Measurement  of  the  Distances  by  Stadia 125 

Exercises  with  the  Plane  Table  : 

132-135.  Practical  Problems 126 

t 

CHAPTER  VI. 

ADDITIONAL  INSTRUMENTS  USED  IN  SURVEYING  AND  PLOTTING. 

The  Aneroid  Barometer; 

136.  The  Aneroid  described 127 

137.  Derivation  of  Barometric  Formulae 129 

138.  Use  of  the  Aneroid 136 

The  Pedometer ; 

139.  The  Pedometer  described 137 

The  Length  of  Men’s  Steps 138 

The  Odometer  : 

140.  Description  and  Use 139 

The  Clinometer  : 

141.  Description  and  Use 141 

The  Optical  Square  : 

142.  Description  and  Use 142 

The  Planimeter : 

143.  Description 143 

144.  Theory  of  the  Polar  Planimeter 144 

145.  To  find  the  Length  of  Arm  to  use 150 

146.  The  Suspended  Planimeter 152 

147.  The  Rolling  Planimeter 152 

148.  Theory  of  the  Rolling  Planimeter  154 

149.  To  Test  the  Accuracy  of  a Planimeter 157 

150.  The  Use  of  the  Planimeter 158 

151.  Accuracy  of  Planimeter  Measurements, 160 

The  Pantograph  : 

152.  Description  and  Theory 161 

Various  Styles  of  Pantographs 163 

153.  Use  of  the  Pantograph 165 


xiv 


CONTENTS. 


PAGR 

Protractors  : 

154.  Various  Styles  described 166 

Parallel  Rulers : 

155.  Description  and  Use 169 

Scales ; 

156.  Various  Kinds  described 169 


BOOK  II. 

SUR  VE  YING  ME  TJ/ODS. 

CHAPTER  VII. 

LAND-SURVEYING. 

157.  Land-surveying  defined 172 

158.  Laying  out  Land 172 

The  United  States  System  of  Laying  out  the  Public  Lands: 

159.  Origin  and  Region  of  Application  of  the  System 173 

160.  The  Reference  Lines 173 

161.  The  Division  into  Townships 174 

162.  The  Division  into  Sections 175 

163.  The  Convergence  of  the  Meridians 176 

164.  The  Corner  Monuments 178 

Finding  the  Area  of  Superficial  Contents  of  Land  when  the 

Limiting  Boundaries  are  given  : 

165.  The  Area  defined 179 

By  Triangular  Subdivision  : 

166.  By  the  Use  of  the  Chain  alone 180 

167.  By  the  Use  of  the  Compass  or  Transit  and  Chain 180 

168.  By  the  Use  of  the  Transit  and  Stadia 181 

From  Bearing  and  Length  of  the  Botmdary  Lines  : 

169.  The  Common  Method 181 

170.  The  Field  Notes 182 

171.  Method  of  Computation  stated 185 

172.  Latitudes,  Departures,  and  Meridian  Distances 1S5 

173.  Computation  of  Latitudes  and  Departures  of  the  Courses 187 

174.  Balancing  the  Survey 190 

175.  The  Error  of  Closure 193 

17C.  The  Form  of  Reduction 194 

177.  Area  Correction  due  to  Erroneous  Length  of  Chain 197 


CONTENTS. 


XV 


PAGE 

Area  from  the  Rectangular  Co  ordinates  of  the  Corners  : 

178.  Conditions  of  Application  of  the  Method. 200 

179.  Theory  of  the  Method 201 

180.  The  Form  of  Reduction 203 

Supplying  Missing  or  Erroneous  Data  : 

181.  Equations  for  Supplying  Missing  Data — Four  Cases 203 

Plotting  : 

i^ia.  Plotting  the  Survey 208 

Irregular  Areas : 

182.  The  Method  by  Offsets  at  Irregular  Intervals 208 

183.  The  Method  by  Offsets  at  Regular  Intervals 210 

The  Subdivision  of  Land  : 

184.  The  Problems  of  Infinite  Variety 213 

185.  To  cut  from  a Given  Tract  of  Land  a Given  Area  by  a Right 

Line  starting  from  a Given  Point  in  the  Boundary 213 

186.  To  cut  from  a Given  Tract  of  Land  a Given  Area  by  a Right  Line 

running  in  a Given  Direction 215 

Examples  : 

187-196^.  Practical  Problems 220-222 

CHAPTER  VIIL 

TOPOGRAPHICAL  SURVEYING  BY  THE  TRANSIT  AND  STADIA. 

197.  Topographical  Survey  defined 223 

198.  Available  Methods 223 

199.  Method  by  Transit  and  Stadia  stated 224 

Theory  of  Stadia  Measurements  : 

200.  Fundamental  Relations 224 

201.  Method  Used  on  the  Government  Surveys 230 

202.  Another  Method  of  Graduating  Rods 231 

203.  Adaptation  of  Formulae  to  Inclined  Sights • 231 

204.  Description  and  Use  of  the  Stadia  Tables 233 

205.  Description  and  Use  of  the  Reduction  Diagram 235 

The  Instruments  : 

206.  The  Transit 235 

207.  Setting  the  Cross-wires 236 

208.  Graduating  the  Stadia  Rod 237 

General  Topographical  Surveying  : 

209.  The  Topography 241 

210.  Methods  of  Field  Work 241 


XVI 


CONTENTS. 


I'ACR 

211.  Reduction  of  the  Notes 249 

212.  Plotting  the  Stadia  Line 252 

213.  Check  Readings 253 

214.  Plotting  the  Side  Readings 254 

215.  Contour  Lines 259 

216.  The  Final  Map 2O2 

217.  Topographical  Symbols 263 

218.  Accuracy  of  the  Stadia  Method  263 

CHAPTER  IX. 

RAILROAD  TOPOGRAPHICAL  SURVEYING. 

219.  Objects  of  the  Survey 265 

220.  The  Field  Work 265 

221.  The  Maps 267 

222.  Plotting  the  Survey 269 

223.  Making  the  Location  on  the  Map 271 

224.  Another  Method 275 

CHAPTER  X. 

HYDROGRAPHIC  SURVEYING. 

225.  Hydrographic  Surveying  defined 277 

The  Location  of  Soundings  ; 

226.  Enumeration  of  Methods 278 

227.  By  Two  Angles  read  on  Shore 279 

228.  By  Two  Angles  read  in  the  Boat — The  Three-point  Problem 279 

229.  By  one  Range  and  one  Angle ' 282 

230.  Buoys,  Buoy-flags,  and  Range-poles 283 

231.  By  one  Range  and  Time-intervals 284 

232.  By  means  of  Intersecting  Ranges 284 

233.  By  Means  of  Cords  or  Wires 284 

Making  the  Soundings  : 

234.  The  Lead 285 

235.  The  Line 285 

236.  Sounding  Poles 287 

237.  Making  Soundings  in  Running  Water 2S7 

238.  The  Water-surface  Plane  of  Reference 2S7 

239.  Lines  of  Equal  Depth 23  • 


CONTENTS. 


XVll 


PAGE 

240.  Soundings  on  Fixed  Cross-sections  in  Rivers 288 

241.  Soundings  for  the  Study  of  Sand-waves 289 

242.  Areas  of  Cross-section 290 

Bench-marks,  Gauges,  Water-levels,  and  Water-Slope  : 

243.  Bench-marks 291 

244.  Water  Gauges 291 

245.  Water-levels 292 

246.  River-slope 293 

The  Discharge  of  Streams  : 

247.  Measuring  Mean  Velocities  of  Water-currents 294 

248.  Use  of  Sub-surface  Floats 295 

249.  Use  of  Current  Meters 300 

250.  Rating  the  Meter 301 

251.  Use  of  Rod  Floats 307 

I 252.  Comparison  of  Methods 308 

253.  The  Relative  Rates  of  Flow  in  Different  Parts  of  the  Cross  section  309 

254.  To  find  the  Mean  Velocity  on  the  Cross-section 312 

255.  Sub-currents 316 

256.  The  Flow  over  Weirs 316 

257.  Weir  Formulae  and  Corrections 319 

258.  The  Miner’s  Inch 322 

259.  Formulae  for  the  Flow  of  Water  in  Open  Channels — Kutter’s  For- 

mula  323 

260.  Cross  sections  of  Least  Resistance 328 

Sediment  Observations  : 

261.  Methods  and  Objects 329 

262.  Collecting  the  Specimens  of  Water 331 

263.  Measuring  out  the  Samples 331 

264.  Siphoning  off,  Filtering,  and  Weighing  the  Sediment 332 

CHAPTER  XL 
MINING  SURVEYING. 

265.  Definitions 333 

266.  Stations 335 

267.  Instruments 335 

268.  Mining  Claims 339 

Underground  Surveys  : 

269.  Mining  Surveying  proper 343 

270.  To  determine  the  Position  of  the  End  or  Breast  of  a Tunnel  and 

its  Depth  below  the  Surface  at  that  Point 343 


XVlll 


CONTENTS. 


PAC.n 

271.  Required,  the  Distance  that  a Tunnel  will  have  to  be  driven  to  cut 

a Vein  with  a Certain  Dip. — Two  Cases 346 

272.  Required  the  Direction  and  Distance  from  the  Dreast  of  a Tunnel 

to  a Shaft,  and  the  Depth  at  which  it  will  cut  the  Shaft 34S 

273.  To  Survey  a ]\Iine  with  its  Shafts  and  Drifts 351 

274.  Conclusion 354 

CHAPTER  XII. 

CITY  SURVEYING. 

275.  Land-surveying  Methods  inadequate  in  City  Work 356 

276.  The  Transit 357 

277.  The  Steel  Tape 357 

Laying  Out  a Town  Site  : 

278.  Provision  for  Growth 359 

279.  Contour  Maps  / 360 

280.  The  Use  of  Angular  Measurements  in  .Subdivisions 360 

281.  Laying  out  the  Ground 361 

2S2.  The  Plat  to  be  Geometrically  consistent 363 

283.  Monuments 363 

284.  Surveys  for  Subdivision 365 

285.  The  Datum-plane 369 

286.  The  Location  of  Streets 369 

287.  Sewer  Systems 370 

2S8.  Water  supply 370 

289.  The  Contour  Map 371 

Methods  of  Measurement  : 

290.  The  Retracing  of  Lines 371 

291.  Erroneous  Standards 372 

292.  True  Standards 373 

293.  The  Use  of  the  Tape 374 

294.  Determination  of  the  “Normal  Tension” 376 

295.  The  Working  Tension 380 

296.  The  Effect  of  Wind 381 

297.  The  Effect  of  Slope. 382 

298.  The  Temperature  Correction 382 

299.  Checks 383 

Miscellaneous  Prohlems  : 

300.  The  Improvement  of  Streets 3^4 

301.  Permanent  Bench-marks 384 


CONTENTS. 


XIX 


302.  The  Value  of  an  Existing  Monument 

303.  The  Significance  of  Possession 

304.  Disturbed  Corners  and  Inconsistent  Plats 

305.  Treatment  of  Surplus  and  Deficiency 

306.  The  Investigation  and  Interpretation  of  Deeds 

307.  Office  Records 

308.  Preservation  of  Lines 

309.  The  Want  of  Agreement  between  Surveyors.. . 


PAGE 

385 

3S7 

38S 

3S9 

391 

391 

392 

393 


CHAPTER  XIII. 

THE  MEASUREMENT  OF  VOLUMES. 


310.  Proposition 394 

311.  Grading  over  Extended  Surfaces 396 

312.  Approximate  Estimates  by  Means  of  Contours 399 

313.  The  Prismoid 402 

314.  The  Prismoidal  Formula 402 

315.  Areas  of  Cross-section 404 

316.  The  Centre  and  Side  Heights 405 

317.  The  Area  of  a Three-level  Section 405 

318.  Cross-sectioning 406 

319.  Three-level  Sections,  the  Upper  Surface  consisting  of  two  Warped 

Surfaces 408 

320.  Construction  of  Tables  for  Prismoidal  Computation • 410 

321.  Three-level  Sections,  the  Surface  divided  into  Four  Planes  by 

Diagonals 413 

322.  Comparison  of  Volumes  by  Diagonals  and  by  Warped  Surfaces. , . 415 

323.  Preliminary  Estimates  from  the  Profiles... 417 

324.  Borrow-pits 420 

325.  Shrinkage  of  Earthwork 420 

326.  Excavations  under  Water 421 


CHAPTER  XIV. 
GEODETIC  SURVEYING. 


527.  Objects  of  a Geodetic  Survey 424 

328.  Triangulation  Systems 425 

329.  The  Base-line  and  its  Connections 427 

330.  The  Reconnaissance  429 


XX 


CONTENTS. 


pAore 

331.  Instrumental  Outfit  for  Reconnaissance 431 

332.  The  Direction  of  Invisible  Stations 432 

333.  The  Heights  of  Stations 432 

334.  Construction  of  Stations 437 

335.  Targets 438 

336.  Heliotropes 442 

337.  Station  Marks 444 

Measuremi-:nt  of  the  Base  Line: 

338.  Methods  447 

The  Steel  Tape 449 

339.  Method  of  Mounting  and  Stretching  the  Tape 450 

340.  M.  Jaderin’s  Method 453 

341.  The  Absolute  Length  of  Tape 455 

342.  The  Coefficient  of  Expansion 456 

343.  The  Modulus  of  Elasticity 457 

344.  Effect  of  the  Sag 457 

345.  Temperature  Correction 459 

346.  Temperature  Correction  when  a Metallic  Thermometer  is  used...  460 

347.  Correction  for  Alignment 461 

348.  Correction  for  Sag 463 

349.  Correction  for  Pull 465 

350.  Elimination  of  Corrections  for  Sag  and  Pull 465 

351.  To  reduce  a Broken  Base  to  a Straight  Line 468 

352.  To  reduce  the  Length  of  the  Base  to  Sea-level 468 

353.  Summary  of  Corrections 469 

354.  To  compute  any  Portion  of  a Broken  Base  which  cannot  be 

directly  measured 472 

355.  Accuracy  attainable  by  Steel  tape  and  Metallic-wire  Measure- 

ments   473 

Measurement  of  the  Angles  : 

356.  The  Instruments 477 

357.  The  Filar  Micrometer 480 

358.  The  Programme  of  Observations 483 

359.  The  Repeating  Method. . . 484 

360.  Method  by  Continuous  Reading  around  the  Horizon 485 

361.  Atmospheric  Conditions 487 

362.  Geodetic  Night  Signals 488 

363.  Reduction  to  the  Centre 488 

Adjustment  of  the  Measured  Angles  : 

364.  Equations  of  Condition 49I 

365.  Adjustment  of  a Triangle 493 


CONTENTS. 


xxi 


PAGE 

Adjustment  of  a Quadrilateral  : 

366.  The  Geometrical  Conditions 494 

367.  The  Angle-equation  Adjustment 495 

368.  The  Side-equation  Adjustment 497 

369.  Rigorous  Adjustment  for  Angle-  and  Side-equations 501 

Example  of  Quadrilateral  Adjustment..^ 504 

Adjustment  of  Larger  Systems  : 

370.  Used  only  in  Primary  Triangulation 506 

371.  Computing  the  Sides  of  the  Triangles 506 

Latitude  and  Azimuth  : 

372.  Conditions 508 

373.  Latitude  and  Azimuth  by  Observations  on  Circumpolar  Stars  at 

Culmination  and  Elongation 508 

374.  The  Observation  for  Latitude 512 

375.  First  Method 513 

376.  Second  Method 513 

377.  Correction  for  Observations  not  on  the  Meridian 514 

378.  The  Observation  for  Azimuth 515 

379.  Corrections  for  Observations  near  Elongation 517 

380.  The  Target 518 

381.  The  Illumination  of  Cross-wires 518 

Time  and  Longitude  : 

382.  Fundamental  Relations 519 

383.  Time 520 

384.  Conversion  of  a Sidereal  into  a Mean  Solar  Time  Interval,  and  vice 

versa 522 

385.  To  change  Mean  Time  into  Sidereal  Time 524 

386.  To  change  from  Sidereal  to  Mean  Time 525 

387.  The  Observation  for  Time 526 

388.  Selection  of  Stars,  with  List  of  Southern  Time-Stars  for  each  Month.  526 

389.  Finding  the  Mean  Time  by  Transit 530 

390.  Finding  the  Altitude 531 

391.  Making  the  Observations 532 

392.  Longitude 534 

393.  Computing  the  Geodetic  Positions 535 

394.  Example  of  ZAf  Z Computation 539 

Geodetic  Levelling  : 

395.  Of  Two  Kinds 340 

(A)  Trigonometrical  Levelling: 

396.  Refraction  540 

397.  Formulae  for  Reciprocal  Observations 541 


XXll 


CONTENTS. 


PAGB 

398.  Formulae  for  Observations  at  One  Station  only 543 

399.  Formulae  for  an  Observed  Angle  of  Depression  to  a Sea  Horizon. . 545 

400.  To  find  the  Value  of  the  Coefficient  of  Refraction 546 

{B)  Precise  Spirit-Levelling; 

401.  Precise  Levelling  defined 547 

402.  The  Instruments 548 

403.  The  Instrumental  Constants 550 

404.  The  Daily  Adjustments 553 

405.  Field  Methods 555 

406.  Limits  of  Error 558 

407.  Adjustment  of  Polygonal  Systems 559 

408.  Determination  of  the  Elevation  of  Mean  Tide 563 

CHAPTER  XV. 

PROJECTION  OF  MAPS,  MAP-LETTERING,  AND  TOPOGRAPHICAL  SYMBOLS. 
Projection  of  Maps: 

409.  Purpose  of  the  Map 564 

410.  Rectangular  Projection 564 

41 1.  Trapezoidal  Projection 565 

412.  The  Simple  Conic  Projection 566 

413.  De  ITsle’s  Conic  Projection 567 

414.  Bonne’s  Projection 567 

415.  The  Polyconic  Projection 568 

416.  Formulae  used  in  the  Projection  of  Maps.  568 

417.  Meridian  Distances  in  Table  VIII 571 

418.  Summary 572 

419.  The  Angle  of  Convergence  of  Meridians 574 

Map-Lettering  and  Topographical  Symbols: 

420.  Map  Lettering 575 

421.  Topographical  Symbols 576 


CONTENTS.  xxiii 


PAGE 

APPENDIX  A. 

The  Judicial  Functions  of  Surveyors 579 

APPENDIX  B. 

Instructions  to  U.  S.  Deputy  Mineral  Surveyors 589 

APPENDIX  C. 

Finite  Differences  605 

APPENDIX  D. 

Derivation  of  Geodetic  Formula 611 


TABLES. 


I. — Trigonometrical  Formula 625 

11. — For  Converting  Metres,  Feet,  and  Chains 629 

III.  — Logarithms  of  Numbers  to  Four  Places 630 

IV. — Logarithmic  Traverse  Table 632 

V. — Stadia  Reductions  for  Horizontal  Distance  and  for  Eleva- 
tion  640 

VI. — Natural  Sines  and  Cosines 648 

^VII. — Natural  Tangents  and  Cotangents 657 

VIII. — Coordinates  for  Polyconic  Projection 669 

IX.— Vai.ues  of  Coefficient  in  Kutter’s  P^ormula  670 

X. — Diameters  of  Circular  Conduits,  by  Kutter’s  Formula 671 

XL — Earthwork  Table — Volumes  by  the  Prismoidal  Formula 672 


SURVEYING 


INTRODUCTION. 

Surveying  is  the  art  of  making  such  field  observations  and 
measurements  as  are  necessary  to  determine  positions,  areas, 
volumes,  or  movements  on  the  earth’s  surface.  The  field  opera- 
tions employed  to  accomplish  any  of  these  ends  constitute  a 
survey.  Accompanying  such  survey  there  is  usually  the  field 
record,  the  computation,  and  the  final  maps,  plats,  profiles,  areas, 
or  volumes.  The  art  of  making  all  these  belongs,  therefore,  to 
the  subject  of  surveying. 

Inasmuch  as  all  fixed  engineering  structures  or  works  involve 
a knowledge  of  that  portion  of  the  earth’s  surface  on  which  they 
are  placed,  together  with  the  necessary  or  resulting  changes  in 
the  same,  so  the  execution  of  such  works  is  usually  accompa- 
nied by  the  surveys  necessary  to  obtain  the  required  informa- 
tion. Thus  surveying  is  seen  to  be  intimately  related  to  en- 
gineering, but  it  should  not  be  confounded  with  it.  All 
engineers  should  have  a thorough  knowledge  of  surveying,  but 
a surveyor  may  or  may  not  have  much  knowledge  of  engineer- 
ing. 

The  subject  of  Surveying  naturally  divides  itself  into — 

I.  The  Adjustment,  Use,  and  Care  of  Instruments. 

II.  Methods  of  Field  Work, 

III.  The  Records,  Computations,  and  Final  Products. 

All  the  ordinary  instruments  that  a surveyor  may  be  called 
upon  to  use  in  any  of  the  departments  of  the  work  will  be  dis- 
cussed in  the  following  pages.  The  most  approved  methods 


2 


INTRODUCTION. 


only  will  be  given  for  obtaining  the  desired  information,  and 
many  problems  that  are  more  curious  than  useful  will  not  be 
mentioned.  The  student  is  assumed  to  possess  a knowledge  of 
geometry,  and  of  plane  and  spherical  trigonometry.  He  is  also 
supposed  to  be  guided  by  an  instructor,  and  have  access  to 
most  of  the  instruments  here  mentioned,  with  the  privilege  of 
using  them  in  the  field. 

The'field  work  of  surveying  consists  wholly  of  measuring  dis- 
tances, angles,  and  time,  and  it  is  well  to  remember  that  no  meas- 
urement can  ever  be  made  exactly.  The  first  thing  the  young  sur- 
veyor needs  to  learn,  therefore,  is  the  proportionate  error 
in  the  special  work  assigned  him  to  perform.  It  is  of  the  utmost 
importance  to  his  success  that  he  shall  thoroughly  study  this 
subject.  He  should  know  what  all  the  sources  of  error  are,  and 
their  relative  importance;  also  the  relative  cost  of  diminishing 
the  size  of  such  errors.  Then,  with  a given  standard  of  accuracy, 
he  will  know  how  to  make  the  survey  of  the  required  standard 
with  the  least  expenditure  of  time  and  labor.  He  must  not  do 
all  parts  of  the  work  as  accurately  as  possible,  or  even  with  the 
same  care.  For,  if  the  expense  is  proportioned  to  the  accuracy 
of  results,  then  he  is  the  most  successful  surveyor  who  does  his 
work  just  good  enough  for  the  purpose.  The  relative  size  of 
the  various  sources  of  error  is  of  the  utmost  importance.  One 
should  not  expend  considerable  time  and  labor  to  reduce  the 
error  of  measurement  of  a line  to  i in  10,000  when  the  unknown 
error  in  the  length  of  the  measuring  unit  may  be  as  high  as  i 
in  1000. 

The  surveyor  must  carefully  discriminate,  also,  between  com- 
pensating errors  and  cumulative  errors.  A compensating  error 
is  one  which  is  as  likely  to  be  plus  as  minus,  and  it  is  therefore 
largely  compensated  in,  or  eliminated  from,  the  result.  A 
cumulative  error  is  one  which  always  enters  with  the  same  sign, 
and  therefore  it  accumulates  in  the  result.  Thus,  in  chaining, 
the  error  in  setting  the  pin  is  a compensating  error,  wliile  the 
error  from  erroneous  length  of  chain  is  a cumulative  error.  If  a 
mile  is  cliained  with  a 66-foot  chain,  tiiere  are  80  measurements 


INTRODUCTION. 


3 


taken.  Suppose  the  error  of  setting  the  pin  be  0.5  inch,  and  the 
error  in  the  length  of  the  chain  be  o.i  inch.  Now  the  theory  of 
probabilities  shows  us  that  in  the  case  of  compensating  errors 
the  square  root  of  tlie  number  of  errors  probably'^  remains  un- 
compensated. The  probable  error  from  setting  the  pins  is 
therefore  9 X 0.5  inch  = 4.5  inches.  The  error  from  erroneous 
length  of  chain  is  80  X o. i inch  = 8 inches.  Thus  we  see  that 
although  the  error  from  setting  the  pins  was  five  times  as  great  as 
that  from  erroneous  length  of  chain,  yet  in  running  one  mile,  the 
resulting  error  from  the  latter  cause  was  nearly  twice  that  from 
the  former.  A careful  study  of  the  various  sources  of  error 
affecting  a given  kind  of  work  will  usually  enable  tlie  surveyor 
either  to  add  greatly  to  its  accuracy  without  increasing  its  cost, 
or  to  greatly  diminish  its  cost  without  diminishing  its  accuracy. 

The  surveyor  should  have  no  desire  except  to  arrive  at  the 
truth.  This  is  the  true  scientific  spirit.  He  should  be  most 
severely  honest  with  himself.  He  should  not  allow  himself 
to  change  or  “fudge”  his  notes  without  sufficient  warrant, 
and  then  a full  explanation  should  be  made  in  his  note-book. 
Neither  should  he  make  his  results  appear  more  accurate  than 
they  really  are.  He  should  always  know  what  was  about  the 
relative  accuracy  witli  which  his  field  work  was  done,  and  carry 
his  results  only  so  far  as  the  accuracy  of  the  work  would  war- 
rant. He  is  either  foolish  or  dishonest  who,  having  made  a 
survey  of  an  area,  for  instance,  with  an  error  of  closure  of  i in 
300,  should  carry  his  results  to  six  significant  figures,  thus  giv- 
ing the  area  to  i in  500,000.  It  is  usual  to  carry  the  computa- 
tions one  place  farther  than  the  results  are  known,  in  order  that 
no  additional  error  may  come  in  from  the  computation.  It  is 
not  unusual,  however,  to  see  results  given  in  published  docu- 
ments to  two,  three,  or  even  four  places  farther  than  the  observa- 
tions would  warrant. 


*The  meaning  of  this  statement  is  that  on  the  average  this  will  occur  oftener 
than  any  other  combination,  and  that  any  single  result  will,  on  the  average,  be 
nearer  to  this  result  than  to  any  other. 


4 


INTRODUCTION. 


The  student  should  make  himself  familiar  witli  the  structure 
and  use  of  every  part  of  every  instrument  put  into  his  hands. 
The  best  way  of  doing  this  is  to  take  the  instrument  all  apart 
and  put  it  together  again.  This,  of  course,  is  not  practicable  for 
each  student  in  college,  but  when  he  is  given  an  instrument  in 
real  practice,  he  should  then  make  himself  thoroughly  familiar 
with  it  before  attempting  to  use  it. 

The  adjustments  of  instruments  should  be  studied  as  problems 
in  descriptive  geometry  and  not  as  mechanical  manipulations, 
learned  in  a mechanical  way;  and  when  adjusting  an  instrument 
the  geometry  of  the  problem  should  be  in  the  mind  rather  than 
the  rule  in  the  memory. 

Students  of  engineering  in  polytechnic  schools  are  urged  to 
make  themselves  familiar  with  every  kind  of  instrument  in  the 
outfit  of  the  institution,  and  to  do  in  the  field  every  kind  of  work 
herein  described  if  pos-sible.  Otherwise  he  may  be  called  upon 
to  do,  or  to  direct  others  to  do,  what  he  has  never  done  himself, 
and  he  will  then  find  that  his  studies  prove  of  little  avail  with- 
out the  real  knowledge  that  comes  only  from  experience. 


BOOK  1. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS. 


CHAPTER  I. 

INSTRUMENTS  FOR  MEASURING  DISTANCES. 

THE  CHAIN. 

1.  The  Engineer’s  Chain  is  50  or  100  feet  long,  and 
should  be  made  of  No.  12  steel  wire.  The  links  are  one  foot 
long,  including  the  connecting  rings, 
and  links  should  be  brazed  to  prevent 
giving.  The  connections  are  designed 
so  as  to  admit  of  as  little  stretch  as 
possible.  Every  tenth  foot  is  marked 
by  a special  form  of  brass  tag.  If  the 
chain  is  adjustable  in  length,  it  should 
be  made  of  standard  length  by  meas- 
uring from  the  inside  of  the  handle  at 
one  end  to  the  outside  of  the  handle 
at  the  other.  If  it  is  not  adjustable, 
measure  from  the  outside  of  the  handle 
at  the  rear  end  to  the  standard  mark 
at  the  forward  end. 

2.  Gunter’s  Chain  is  66  feet  long,  and  is  divided  into  100 
links,  each  link  being  7.92  inches  in  length.  This  chain  is 
mostly  used  in  land-surveying,  where  the  acre  is  the  unit  of 
measure.  It  was  invented  by  Edmund  Gunter,  an  English 


All  joints  in  rings 


Fig.  I. 


6 


SURVEYING. 


astronomer,  about  1620,  and  is  very  convenient  for  obtaining 
areas  in  acres  or  distances  in  miles.  Thus, 

One  mile  = 80  chains  ; also, 

One  acre  = 160  square  rods, 

= 10  square  chains, 

= 100,000  square  links. 

If,  therefore,  the  unit  of  measure  be  chains  and  hundredths 
(links),  the  area  is  obtained  in  square  chains  and  decimals,  and 
by  pointing  off  one  more  place  the  result  is  obtained  in  acres. 
This  is  the  length  of  chain  used  on  all  the  U.  S.  land  surveys. 
In  all  deeds  of  conveyance  and  other  documents,  when  the 
word  chain  is  used  it  is  Gunter’s  chain  that  is  meant. 

3.  Testing  the  Chain. — No  chain,  of  whatever  material 
or  manufacture,  will  remain  of  constant  length.  The  length 
changes  from  temperature,  wear,  and  various  kinds  of  distor- 
tion. A change  of  temperature  of  70°  F.  in  a lOO-foot  chain 
will  change  its  length  by  0.05  foot,  or  a change  of  i in  2000. 

If  the  links  of  a chain  are  joined  by  three  rings,  then  there 
are  eight  wearing  surfaces  for  each  link,  or  eight  hundred 
wearing  surfaces  for  a 66-  or  loo-foot  chain.  If  each  surface 
should  wear  o.oi  inch,  the  chain  is  lengthened  by  eight  inches. 
It  is  not  uncommon  for  a railroad  survey  of,  say,  300  miles  to 
be  run  with  a single  chain.  If  such  a chain  were  of  exactly  the 
right  length  at  the  beginning  of  the  survey,  it  might  be  six 
inches  too  long  at  the  end  of  it. 

The  change  of  length  from  distortion  may  come  from  a 
flattening  out  of  the  connecting  rings,  from  bending  the  links, 
or  from  stretching  the  chain  beyond  its  elastic  limit,  thus  giv- 
ing it  a permanent  set.  Both  the  wear  and  the  distortion  are 
likely  to  be  less  for  a steel  chain  than  for  an  iron  one.  When 
a bent  link  is  straightened  it  is  permanently  lengthened. 

When  we  remember  that  all  unknowm  changes  in  the 
length  of  the  chain  produce  cumulative  errors  in  the  meas- 
ured lines,  we  see  how  important  it  is  that  the  true  length  of 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS,  / 


the  chain  should  be  always  known,  or  better,  that  the  standard 
length  (50,  66,  or  100  feet)  should  be  properly  measured  from 
one  end  of  the  chain  and  marked  at  the  other.  This  chain 
test  is  most  readily  accomplished  by  the  aid  of  a standard  steel 
tape,  which  is  at  least  as  long  as  the  chain.  By  the  aid  of  such 
a tape  a standard  length  may  be  laid  off  on  the  floor  of  a large 
room,  or  two  stones  may  be  firmly  set  in  the  ground  at  the 
proper  distance  apart  and  marks  cut  upon  their  upper  sur- 
faces. If  stones  are  used  they  should  reach  below  the  frost- 
line. Or  a short  tape,  or  other  standard  measuring  unit,  may 
be  used  for  laying  off  such  a base-line.  By  whatever  means  it 
is  accomplished,  some  ready  means  should  at  all  times  be 
available  for  testing  the  chain.  Since  a chain  always  grows 
longer  with  use,  the  forward  end  of  the  chain  will  move 
farther  and  farther  from  the  standard  mark.  A small  file- 
mark  may  be  made  on  the  handle  or  elsewhere,  and  then  re- 
moved when  a new  test  gives  a new  position.  Care  must  be 
exercised  to  see  that  there  are  no  kinks  in  the  chain  either  in 
testing  or  in  use. 

In  laying  out  the  standard  base  the  temperature  at  which 
the  unit  of  measure  is  standard  should  be  known  (this  tempera- 
ture is  stamped  on  the  better  class  of  steel  tapes),  and  if  the 
base  is  not  laid  out  at  this  temperature,  a correction  should 
be  made  before  the  marks  are  set.  The  coefficient  of  expansion 
of  iron  and  steel  is  very  nearly  0.0000065  for  1°  F.  If  T'o  be 
the  temperature  at  which  the  tape  is  standard,  T the  tem- 
perature at  which  the  base  is  measured,  and  L the  length 
of  the  base,  then  0.0000065  {T^—  T)L  is  the  correction  to  be 
applied  to  the  measured  length  to  give  the  true  length. 

When  the  chain  is  tested  by  this  standard  base  the  tem- 
perature should  be  again  noted,  and  if  this  is  about  the  mean 
temperature  for  the  field  measurements  no  correction  need  be 
made  to  the  field  work.  If  it  is  known,  at  the  time  the  chain 
is  tested,  that  the  temperature  is  very  different  from  the  prob- 


8 


S UR  VE  YING. 


able  mean  of  the  field  work,  then  the  standard  mark  can  be  so 
placed  on  the  chain  as  to  make  it  standard  when  in  use. 

4.  The  Use  of  the  Chain. — The  chain  is  folded  by  taking 
it  by  the  middle  joint  and  folding  the  two  ends  simultaneous- 
ly. It  is  opened  by  taking  the  two  handles  in  one  hand  and 
throwing  the  chain  out  with  the  other. 

Since  horizontal  distances  are  always  desired  in  surveying, 
the  chain  should  be  held  horizontally  in  measuring.  Points 
vertically  below  the  ends  of  the  chain  are  marked  by  iron  pins, 
the  head  chainman  placing  them  and  the  rear  chainman  remov- 
ing them  after  the  next  pin  is  set.  The  chain  is  lined  in  either 
by  the  head  or  rear  chainman,  or  by  the  observer  at  the  instru- 
ment, according  as  the  range-pole  is  in  the  rear,  or  in  front,  or 
not  visible  by  either  chainman.  When  chaining  on  level 
ground,  the  rear  chainman  brings  the  outside  of  the  handle 
against  the  pin,  and  the  head  chainman  sets  the  forward  side 
of  his  pin  even  with  the  standard  mark  on  the  chain.  By  this 
means  the  centres  of  the  pins  are  the  true  distance  apart.  On 
uneven  ground  both  chainmen  cannot  hold  to  the  pin  ; one  end 
being  elevated  in  order  to  bring  the  chain  to  a horizontal 
position.  In  this  case  there  are  three  difficulties  to  be  over- 
come. The  chain  should  be  drawn  so  taut  that  the  stretch 
from  the  pull  would  balance  the  shortening  from  the  sag;  the 
chain  should  be  made  horizontal ; the  elevated  end-mark  must 
be  transferred  vertically  to  the  ground.  It  is  practically  im- 
possible to  do  any  of  these  exactly.  The  first  could  be  deter- 
mined by  trial.  Stretch  the  chain  between  two  points  at  the 
same  elevation,  having  it  supported  its  entire  length.  Then 
remove  the  supports,  and  see  how  strong  a pull  is  required  to 
bring  it  to  the  marks  again.  This  should  be  done  by  the  chain- 
men  themselves,  thus  enabling  them  to  judge  how  hard  to  pull 
it  when  it  is  off  the  ground.  To  hold  the  chain  horizontal  on 
sloping  ground  is  very  difficult,  on  account  of  the  judgment 
being  usually  very  much  in  error  as  to  the  position  of  a hori- 


ADJUSTMENT,  USE,  AND  CADE  OF  INSTRUMENTS.  9 


zontal  line.  In  all  such  cases  the  apparently  horizontal  line  is 
much  too  nearly  parallel  with  the  ground.  Sometimes  a level  has 
been  attached  to  one  end  of  the  chain,  in  which  case  it  should 
be  adjusted  to  indicate  horizontal  end-positions  for  a certain 
pull,  this  being  the  pull  necessary  to  overcome  the  shortening 
from  sag.  To  hold  a plumb-line  at  the  proper  mark,  with  the 
chain  at  the  right  elevation,  and  stretched  the  proper  amount, 
requires  a steady  hand  in  order  that  the  plumb-bob  may  hang 
stationary.  This  should  be  near  the  ground,  and  when  all  is 
ready,  it  is  dropped  by  the  chainman  letting  go  the  string. 
The  pin  is  then  stuck  and  the  work  proceeds.  It  is  common 
in  this  country  for  the  rear  chainman  to  call  “ stick”  when  he 
is  ready,  and  for  the  head  chainman  to  answer  “ stuck”  when 
he  has  set  the  pin.  The  rear  chainman  then  pulls  his  pin  and 
walks  on. 

There  should  be  eleven  pins,  marked  with  strips  of  colored 
flannel  tied  in  the  rings  to  assist  in  finding  them  in  grass  or 
brush.  In  starting,  the  rear  chainman  takes  a pin  for  the  initial 
point,  leaving  the  head  chainman  with  ten  pins.  When  the 
last  pin  is  stuck,  the  head  chainman  calls  out,”  and  waits  by 
this  station  until  the  rear  chainman  comes  up  and  delivers  over 
the  ten  pins  now  in  his  possession.  The  eleventh  pin  is  in  the 
ground,  and  serves  as  the  initial  point  for  the  second  score. 
Thus  only  every  ten  chains  need  be  scored. 

Good  chaining,  therefore,  consists  in  knowing  the  length  of 
the  chain,  in  true  alignment,  horizontal  and  vertical,  and  in 
proper  stretching,  marking,  and  scoring. 

THE  STEEL  TAPE.- 

5.  Varieties. — Steel  tapes  are  now  made  from  one  yard  to 
1000  feet  in  length,  graduated  metrically,  or  in  feet  and  tenths. 
A pocket  steel  tape  from  three  to  ten  feet  long  should  always 
be  carried  by  the  surveyor.  A 50-foot  tape  is  best  fitted  to 
city  surveying  where  there  are  appreciable  grades.  For  cities 


lO 


SU/^  VE  YING. 


without  grades  a loo-foot  tape  might  be  found  more  useful. 
For  measuring  base-lines,  or  for  some  kinds  of  mining  surveying, 
a 300  or  500  foot  tape  is  best.  These  are  of  small  cross-section, 
being  about  o.i  inch  wide  and  0.02  inch  thick.  A tape  about 


Fig.  2. 


0.5  inch  wide  and  0.02  fneh  thick  (Fig.  2)  is  perhaps  best  suited 
to  general  surveying. 

6.  The  Use  of  Steel  Tapes. — Steel  tape-measures  are  used 
just  as  chains  are.  They  are  provided  with  handles,  but  the 
end  graduation-marks  are  usually  on  the  tape  itself  and  not  on 
the  handle.  They  are  graduated  to  order,  the  graduations 
being  either  etched  or  made  on  brass  sleeves  which  are  fastened 
on  the  tape.  Their  advantages  are  many.  They  do  not  kink, 
stretch,  or  wear  so  as  to  change  their  length,  so  that,  with 
careful  handling,  they  remain  of  constant  length  except  for 
temperature.  They  are  used  almost  exclusively  in  city  and 
bridge  work,  and  in  the  measurement  of  secondary  base-lines. 
The  same  precautions  must  be  taken  in  regard  to  alignment, 
pull,  and  marking  with  the  tape,  as  was  described  for  the 
chain.'^* 


* For  methods  of  using  the  steel  tape  in  accurate  measurements,  see  Chap- 
ter XIV.,  Base-Line  Measurements. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  II 


EXERCISES. 

To  be  worked  out  on  the  ground  by  the  use  of  the  chain  or  tape  alone. 

7.  To  chain  a line  over  a hill  between  two  given  points,  not  visible  from 
each  other. 

Range-poles  are  set  at  the  given  points.  Then  the  two  chainmen,  each  with 
a range-pole,  range  themselves  in  between  the  two  fixed  points,  near  the  top 
of  the  hill,  by  successive  approximations.  The  line  can  then  be  chained. 

8.  To  chain  a line  across  a valley  between  two  fixed  points. 

Establish  other  range-poles  by  means  of  a plumb-line  held  on  range  between 
the  points. 

9.  To  chain  a line  between  two  fixed  points  when  woods  intervene,  and  the 
true  line  is  not  to  be  cleared  out. 

Range  out  a trial  line  by  poles,  leaving  fixed  points.  Find  the  resulting  error 
at  the  terminus,  and  move  all  the  points  over  their  proportionate  amount.  The 
true  line  may  then  be  chained. 

10.  To  set  a stake  in  a line  perpendicular  to  a given  line  at  a given  point. 

All  multiples  of  3,  4,  and  5 are  the  sides  of  a right-angled  triangle;  also  any 

angle  in  a semicircumference  is  a right  angle. 

11.  To  find  where  a perpendicular  from  a given  point  without  a line  will  meet 
that  line. 

Run  an  inclined  line  from  the  given  point  to  the  given  line.  Erect  a per- 
pendicular from  the  given  line  near  the  required  point,  extend  it  till  it  intersects 
the  inclined  line,  and  solve  by  similar  triangles. 

12.  To  establish  a second  point  that  shall  make  with  a given  point  a line 
parallel  to  a given  line. 

Diagonals  of  a parallelogram  bisect  each  other. 

13.  To  determine  the  horizontal  distance  from  a given  point  to  a visible  but 
inaccessible  object. 

Use  two  similar  right-angled  triangles. 

14.  To  prolong  a line  beyond  an  obstacle?  in  azimuth*  and  distance. 

First  Solution  : By  an  equilateral  triangle. 

Second  Solution  : By  two  rectangular  offsets  on  each  side  of  the  obstacle. 

Third  Solution  : By  similar  triangles,  as  in  Fig.  3. 

From  any  point  as  A run  the  line  AB,  fixing  the  half  and  three  quarter  points 
at  X and  j.  From  any  other  point  as  C,  run  CxD,  making  xD  = Cx.  From  D 


*The  azimuth  of  a line  is  the  angle  it  forms  with  the  meridian,  and  is  meas- 
ured from  the  south  point  in  the  direction  S.W.  N.E.  to  360  degrees.  It  thus 
becomes  a definite  direction  when  the  angle  alone  is  given.  Thus  the  azimuth 
of  220'’  corresponds  to  the  compass-bearing  of  N.  40°  E. 


12 


SUR  VE  YING. 


run  DyE  making  DE  = AB=  4Z?/,  fixing  the  middle  point  z.  From  B run 
BzII,  making  zH  = Bz.  Then  is  HE  parallel  and  equal  to  DB,  A C,  and  CH. 


D B 

Fig.  3. 


Stakes  should  be  set  at  all  the  points  lettered  in  the  figure.  Check:  Measure 
HE  and  AC.  If  they  are  equal  the  work  is  correct. 

15.  To  measure  a given  angle. 

Lay  off  equal  distances,  b,  from  the  vertex  on  the  two  lines,  and  measure  the 

a 

third  side  a of  the  triangle.  Then  tan  ^ A=  --  ■ -r— 

V4b'^  — a* 

16.  To  lay  out  a given  angle  on  the  ground. 

Reverse  the  above  operation.  .<4  is  known;  assume  ^ and  compute  a.  Then 
from  A measure  oft  AB  = b.  From  B and  A strike  arcs  with  radii  equal  to  a 
and  b respectively,  giving  an  intersection  at  C.  Then  CAB  is  the  required 
angle.  If  b is  assumed  not  greater  than  0.6  the  length  of  the  chain,  angles  may 
be  laid  out  up  to  go®. 


17.  Other  Instruments  for  measuring  distances  with  great 
accuracy  will  be  discussed  under  the  head  of  Base-Line 
Measurements,  Chapter  XIV. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  1 3 


CHAPTER  II. 

INSTRUMENTS  FOR  DETERMINING  DIRECTIONS. 

THE  COMPASS. 

l8.  The  Surveyor’s  Compass  consists  essentially  of  a line 
of  sight  attached  to  a horizontal  graduated  circle,  at  the  centre 
of  which  is  suspended  a magnetic  needle  free  to  move,  the 
whole  conveniently  supported  with  devices  for  levelling.  Fig. 


4 shows  a very  good  form  of  such  an  instrument.  In  ad- 
dition to  the  above  essential  features,  the  instrument  here 
shown  has  a tangent-screw  and  vernier-scale  at  e for  setting 
off  the  declination  of  the  needle;  a tangent-scale  on  the  edge 
of  the  vertical  sight  for  reading  vertical  angles,  the  eye  being 
placed  at  the  sight-disk  shown  on  the  opposite  standard ; and  an 


H 


SUR  VE  YTNG. 


auxiliary  graduated  circle,  with  vernier,  shown  on  the  front 
part  of  the  plate,  for  reading  angles  closer  than  could  be  done 
with  the  needle.  The  compass  is  mounted  either  on  a tripod 
or  on  a single  support  called  a Jacob’s-staff.  It  is  connected 
to  its  support  by  a ball-and-socket  joint,  which  furnishes  a con- 
venient means  of  levelling. 

Although  the  needle-compass  does  not  give  very  accurate 
results,  it  is  one  of  the  most  useful  of  surveying -instruments. 
Its  great  utility  lies  in  the  fact  that  the  needle  always  points 
in  a known  direction,  and  therefore  the  direction  of  any  line 
of  sight  may  be  determined  by  referring  it  to  the  needle-bear- 
ing. The  needle  points  north  in  only  a few  localities;  but  its 
declination  from  the  north  point  is  readily  determined  for  any 
region,  and  then  the  true  azimuth,  or  bearing  of  a line,  may  be 
found.  It  has  grown  to  be  the  universal  custom,  in  finding 
the  direction  of  a line  by  the  compass,  to  refer  it  to  cither 
the  north  or  the  south  point,  according  to  which  one  gives  an 
acute  angle.  Thus,  if  the  bearing  is  ioo°  from  the  south 
point  it  is  but  8o°  from  the  north  point,  and  the  direction 
would  be  defined  as  north,  8o°  east  or  west,  as  the  case 
might  be:  thus  no  line  can  have  a numerical  bearing  of 
more  than  90°.  In  accordance  with  this  custom,  all  needle- 
compasses  are  graduated  from  both  north  and  south  points 
each  way  to  the  east  and  west  points,  the  north  and  south 
points  being  marked  zero,  and  the  east  and  west  points  90°. 
When  the  direction  of  a line  is  given  by  this  system  it  is 
called  the  bearing  of  the  line.  When  it  is  simply  referred 
to  the  position  of  the  needle  it  is  called  the  magnetic  bearing. 
When  it  is  corrected  for  the  declination  of  the  needle, 
either  by  setting  off  the  declination  on  the  declination-arc  or 
by  correcting  the  observed  reading,  it  is  called  the  true  bear- 
ing, being  then  referred  to  the  true  meridian. 

Becau.se  the  graduated  circle  is  attached  to  the  line  of  sight 
and  moves  with  it,  while  the  needle  remains  stationary,  E and 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  1 5 


W are  placed  on  the  compass-circle  in  reversed  position. 
Thus  when  the  line  of  sight  is  north-east,  the  north  end  of  the 
needle  points  to  the  left  of  the  north  point  on  the  circle,  and 
hence  E must  be  put  on  this  side  of  the  meridian  line. 

In  reading  the  compass,  always  keep  the  north  end  of  the  circle 
pointing  forward  alo7tg  the  line,  and  read  the  north  end  of  the 
needle. 

The  north  end  of  the  needle  is  usually  shaped  to  a special 
design,  or,  if  not,  it  may  be  distinguished  by  knowing  that  the 
south  end  is  weighted  by  having  a small  adjustable  brass  wire 
slipped  upon  it  to  overcome  the  tendency  the  north  end  has 
to  dip. 


ADJUSTMENTS  OF  THE  COMPASS. 

19.  The  General  Principle  of  almost  all  instrumental  ad- 
justments is  the  Principle  of  Reversion,  whereby  the  error  is 
doubled  and  at  the  same  time  made  apparent.  A thorough  mas- 
tery of  this  principle  will  nearly  always  enable  one  to  deter- 
mine the  proper  method  of  adjusting  all  parts  of  any  survey- 
ing instrument.  It  should  be  a recognized  principle  in  sur- 
veying, that  no  one  is  competent  to  handle  any  instrument 
who  is  not  able  to  determine  when  it  is  in  exact  adjustment, 
to  locate  the  source  of  the  error  if  not  in  adjustment,  to  dis- 
cuss the  effect  of  any  error  of  adjustment  on  the  work  in 
hand,  and  to  properly  adjust  all  the  movable  parts.  The 
methods  of  adjustment  should  not  be  committed  to  memory — 
any  more  than  should  the  demonstration  of  a proposition  in 
geometry.  The  student  in  reading  the  methods  of  adjust- 
ment should  see  that  they  are  correct,  just  as  he  sees  the  cor- 
rectness of  a geometrical  demonstration.  Having  thus  had 
the  method  and  the  reason  therefor  clearly  in  the  mind,  he 
should  trust  his  ability  to  evolve  it  again  whenever  called 
upon.  He  thus  relies  upon  the  accuracy  of  his  reasoning, 
rather  than  on  the  distinctness  of  his  recollection. 


i6 


S UR  VE  YING. 


20.  To  make  the  Plate  perpendicular  to  the  Axis  of  the 
Socket. — This  must  be  done  by  the  maker.  It  is  here  men- 
tioned because  the  axis  is  so  likely  to  get  accidentally  bent. 
Instruments  made  of  soft  brass  must  be  handled  very  care- 
fully to  prevent  such  an  accident.  If  this  adjustment  is  found 
to  be  very  much  out,  it  should  be  sent  to  the  makers.  If 
much  out,  it  will  be  shown  by  the  needle,  and  also  by  the 
plate-bubbles. 

21.  To  make  the  Plane  of  the  Bubbles  perpendicular 
to  the  Axis  of  the  Socket. — Level  it  in  one  position,  turn 
i8o°,  and  correct  one  half  the  movement  of  each  bubble  by 
the  adjusting-screw  at  the  end  of  the  bubble-case.  Now  level 
up  again,  and  revolve  i8o°,  and  the  bubbles  should  remain  at 
the  centre.  If  not,  adjust  for  one  half  the  movement  again, 
and  so  continue  until  the  bubbles  remain  in  the  centre  for  all 
positions  of  the  plate. 

The  student  should  construct  a figure  to  illustrate  this  and  almost  all  other 
adjustments.  Thus,  in  this  case,  let  the  figure  consist  of  two  lines,  one  repre- 
senting the  axis  of  the  socket,  and  the  other  the  axis  of  the  bubble,  crossing  it. 
Now  if  these  two  lines  are  not  at  right  angles  to  each  other,  when  the  one  is 
horizontal  (as  the  bubble-axis  is  when  the  bubble  rests  at  the  centre  of  its  tube) 
the  other  is  inclined  from  the  vertical.  Now  with  this  latter  fixed,  let  the 
figure  be  revolved  i8o°  about  it  (or  construct  another  figure  representing  such 
a movement),  and  it  will  be  seen  that  the  bubble-axis  now  deviates  from  the 
horizontal  by  twice  the  difference  between  the  angle  of  the  lines  and  90°,  By 
now  correcting  o}ie  half  of  this  change  of  direction  on  the  part  of  the  bubble- 
axis,  it  will  be  made  perpendicular  to  the  socket-axis.  Then  by  relevelling  the 
instrument,  which  consists  of  moving  the  socket-axis  until  the  bubbles  return 
to  the  middle  of  the  tubes,  the  instrument  should  now  revolve  in  a horizontal 
plane. 


22.  To  adjust  the  Pivot  to  the  Centre  of  the  Graduated 
Circle. — When  the  two  ends  of  the  needle  do  not  read  exactly 
alike  it  may  be  due  to  one  or  more  of  three  causes:  The 
circle  may  not  be  uniformly  graduated  ; the  pivot  may  be  bent 
out  of  its  central  position ; or  the  needle  may  be  bent.  All 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS,  1 7 


our  modern  instruments  are  graduated  by  machinery,  so  that 
they  have  no  errors  of  graduation  that  could  be  detected  by 
eye.  One  or  both  of  the  other  two  causes  must  therefore  ex- 
ist. If  the  difference  between  the  two  end-readings  is  con- 
stant for  all  positions  of  the  needle,  then  the  pivot  is  in  the 
centre  of  the  circle,  but  the  needle  is  bent.  If  the  difference 
between  the  two  end-readings  is  variable  for  different  parts  of 
the  circle,  then  the  pivot  is  bent,  and  the  needle  may  or  may 
not  be  straight.  To  adjust  the  pivot,  therefore,  find  the  posi- 
tion of  the  needle  which  gives  the  maximum  difference  of  end- 
readings,  remove  the  needle,  and  bend  the  pivot  at  right  angles 
to  this  position  by  one  half  the  difference  in  the  extreme  variation 
of  end-readings.  Repeat  the  test,  etc.  Since  the  glass  cover 
is  removed  from  the  compass-box  in  making  this  adjustment, 
it  should  be  made  indoors,  to  prevent  any  disturbance  from 
wind. 

23.  To  straighten  the  Needle,  set  the  north  end  exactly 
at  some  graduation-mark,  and  read  the  south  end.  If  not  180° 
apart,  bend  the  needle  until  they  are.  This  implies  that  the 
preceding  adjustment  has  been  made,  or  examined  and  found 
correct. 

24.  To  make  the  Plane  of  the  Sights  normal  to  the 
Plane  of  the  Bubbles. — Carefully  lewel  the  instrument  and 
bring  the  plane  of  the  sights  upon  a suspended  plumb-line. 
If  this  seems  to  traverse  the  farther  slit,  then  that  sight  is  in 
adjustment.  Reverse  the  compass,  and  test  the  other  sight 
in  like  manner.  If  either  be  in  error,  its  base  must  be  re- 
shaped to  make  it  vertical. 

25.  To  make  the  Diameter  through  the  Zero-gradua- 
tions lie  in  the  Plane  of  the  Sights. — This  should  be  done  by 
the  maker,  but  it  can  be  tested  by  stretching  two  fine  hairs 
vertically  in  the  centres  of  the  slits.  The  two  hairs  and  the 
two  zero-graduations  should  then  be  seen  to  lie  in  the  same 
plane.  The  declination-arc  must  be  set  to  read  zero. 


i8 


SURVEYING. 


26.  To  remagnetize  the  Needle. — Needles  sometimes  lose 
their  magnetic  properties.  They  must  then  be  remagnetized. 
To  do  this  take  a simple  bar-magnet  and  rub  each  end  of  the 
needle,  from  centre  towards  the  ends,  with  the  end  of  the 
magnet  which  attracts  in  each  case.  In  returning  the  magnet 
for  the  next  stroke  lift  it  up  a foot  or  so  to  remove  it  from 
the  immediate  magnetic  field,  otherwise  it  would  tend  to  mil- 
lify  its  own  action.  The  needle  should  be  removed  from  the 
pivot  in  this  operation,  and  the  work  continued  until  it  shows 
due  activity  when  suspended.  An  apparently  sluggish  needle 
may  be  due  to  a blunt  pivot.  If  so,  this  should  be  removed, 
and  ground  down  on  an  oil-stone. 

THE  VERNIER. 

27.  The  Vernier  is  an  auxiliary  scale  used  for  reading  frac- 
tional parts  of  the  divisions  on  the  main  graduated  scale  or  limb. 
If  we  wish  to  read  to  tenths  of  one  division  on  the  limb,  we 
make  10  divisions  on  the  vernier  correspond  to  either  9 or  1 1 
divisions  on  the  limb.  Then  each  division  on  the  vernier  is 
one  tenth  less  or  greater  than  a division  on  the  limb.  If  we 
wish  to  read  to  twentieths  or  thirtieths  of  one  division  on  the 
limb,  there  must  be  twenty  or  thirty  divisions  on  the  vernier 
corresponding  to  one  n^ore  or  less  on  the  limb. 

The  zero  of  the  vernier-scale  marks  the  point  on  the  limh 
whose  reading  is  desired. 

Suppose  this  zero-point  corresponds  exactly  with  a division 
on  the  limb.  The  reading  is  then  made  wholly  on  the  limb. 
If  a division  on  the  vernier  is  less  than  a division  on  the  limb, 
then,  by  moving  the  forward  a trifle,  the  next  fo7'ward 

division  on  the  vernier  corresponds  with  a division  on  the  limb. 
(The  particular  division  on  the  limb  that  may  be  in  coincidence 
is  of  no  consequence.)  On  the  other  hand,  if  a division  on  the 
vernier  greater  than  a division  on  the  limb,  then  by  moving 
the  vernier  forward  a trifle,  the  next  backward  division  on  the 


ADJUSTMENT,  USE,  AND  CAEE  OF  INSTRUMENTS.  1 9 


vernier  comes  into  coincidence.  Thus  we  have  two  kinds  of 
verniers,  direct  and  retrograde  according  as  they  are  read 

forward  or  backward  from  the  zero-point.  Most  verniers  in 
use  are  of  the  direct  kind,  but  those  commonly  found  on  sur- 
veyors’ compasses  for  setting  off  the  declination  are  generally 
of  the  retrograde  order. 


In  Fig.  5 are  shown  two  direct  verniers,  such  as  are  used 
on  transits  with  double  graduations.  Thus  in  reading  to  the 
right  the  reading  is  138°  45',  but  in  reading  to  the  left  it  is  221° 
15'.  In  each  case  we  look  along  the  vernier  in  the  direction  of 
the  graduation  for  the  coincident  lines. 


In  Fig.  6 is  shown  a special  form  of  retrograde  vernier  in 
which  the  same  set  of  graduation-lines  on  the  vernier  serve  for 


20 


SURVEYING. 


either  right-  or  left-hand  angles.  Here  a division  of  the  vernier 
is  larger  than  a division  on  the  limb,  and  it  must  therefore  be 
read  backwards.  Thus,  we  see  that  the  zero  of  the  vernier 
is  to  the  left  of  the  zero  of  the  limb,  the  angle  being  30'  and 
something  more.  Starting  now  toward  the  right  (backwards) 
on  the  vernier  scale,  we  reach  the  end  or  15-minute  mark, 
without  finding  coincident  lines  ; we  then  skip  to  the  left-hand 
side  of  the  vernier  scale  and  proceed  iozvards  the  right  again 
until  we  find  coincident  lines  at  the  twenty-sixth  mark.  The 
reading  is  therefore  ^o-\-26=^6  minutes.  This  is  the  form 
of  vernier  usually  found  on  surveyors’  compasses  for  setting 
off  the  declination.  We  have  therefore  the  following 


Rules. 

First.  To  find  the  ‘‘  smallest  reading"  of  the  vernier.,  divide 
the  value  of  a division  on  the  limb  by  the  number  of  divisions  in 
the  vcriiier. 

Second.  Read  forward  along  the  limb  to  the  last  graduation 
precedmg  the  zero  of  the  vernier ; then  read  forward  along  the 
vernier  if  direct^  or  backward  if  retrograde,  until  coincident  lines 
are  found.  The  number  of  this  line  on  the  vernier  from  the  zero- 
graduation  is  the  number  of  smallest-reading"  units  to  be 
added  to  the  reading  made  on  the  hmb. 

These  rules  apply  to  all  verniers,  whether  linear  or  circular. 

THE  DECLINATION  OF  THE  NEEDLE. 

28.  The  Declination^  of  the  Needle  is  the  horizontal 
angle  it  makes  with  the  true  meridian.  At  no  place  on  the 
earth  is  this  angle  a constant.  The  change  in  this  angle  is 
called  the  variation  of  the  declination. 

29.  The  Daily  Variation  in  the  Declination  consists  in  a 

Formerly  called  variation  of  the  needle,  and  still  so  called  by  navigators 
and  by  many  surveyors. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  2 1 


swinging  of  the  needle  through  an  arc  of  about  eight,  minutes 
daily,  the  north  end  having  its  extreme  easterly  variation  about 
8 A.M.  and  its  extreme  westerly  position  about  p.30  P.M.  It 
has  its  mean  or  triLe  declination  about  10.30  A.M.  and  8 P.M. 
It  varies  with  the  latitude  and  with  the  season,  but  the  foITow- 
ing  table  gives  a fair  average  for  the  United  States.  A more 
extended  table  may  be  found  in  the  Report  of  the  U.  S.  Coast 
and  Geodetic  Survey  for  1881,  Appendix  8. 


TABLE  OF  CORRECTIONS  TO  REDUCE  OBSERVED  BEARINGS 
TO  THE  DAILY  MEAN. 


Month. 

Add  to  N.E.  and  S.W. 
bearings. 

Subtract  from  N.W.  and 
S.E.  bearings. 

Add  to  N.W.  and  S.E.  bearings. 
Subtract  from  N.E.  and  S.W.  bearings. 

6 

7 

8 

9 

10 

II 

12 

I 

2 

3 

4 

5 

6 

A.M. 

A.M. 

A.M. 

A.M. 

A.M. 

A.M. 

M. 

P.M. 

P.M. 

P.M. 

P.M, 

P.M, 

P.M. 

January  

l' 

2' 

2' 

l' 

0' 

2' 

3' 

3' 

2' 

l' 

l' 

o' 

April 

3 

4 

4 

3 

I 

I 

4 

5 

5 

4 

3 

2 

I 

July 

4 

5 

5 

4 

I 

I 

4 

5 

5 

4 

3 

2 

I 

October 

2 

2 

2 

I 

I 

3 

3 

3 

2 

I 

0 

0 

This  table  is  correct  to  the  nearest  minute  for  Philadelphia,  where  the  observations  were 

made. 


30.  The  Secular  Variation  of  the  magnetic  declination  is 
probably  of  a periodic  character,  requiring  two  or  three  cen- 
turies to  complete  a cycle.  The  most  extensive  set  of  obser- 
vations bearing  on  this  subject  have  been  made  at  Paris,  where 
records  of  the  magnetic  declination  have  been  kept  for  about 
three  and  a half  centuries.  The  secular  variation  for  Paris  is 
shown  in  Fig.  7,  and  that  for  Baltimore,  Md.,  in  Fig.  8.* 

Whether  or  not  either  of  these  curves  will  return  in  time  to 
the  same  extreme  limits  here  given  is  unknown,  as  is  also  the 
cause  of  these  remarkable  changes.  The  extraordinary  varia- 
tion in  the  declination  at  Paris  of  some  32°,  and  that  at 


* These  taken  from  the  Coast  Survey  Report  of  1882. 


22 


SURVEYING. 


Baltimore  of  some  5°,  show  the  necessity  of  paying  careful 
attention  to  this  matter.  No  reliance  should  be  placed  on 


1540 

CO 

80 

ICOO 

20 

40 

GO 

80 

1700 

20 

40 

CO 

80 

1800 

20 

40 

60 

'80 

1900. 


MM 

_ i n~n 

M 1 1 1 1 

1 

1 

Secular 

Vdriationl 

1 df,ilie  Magnetic  Declination  at  Pay 

is 

Fra 

Tice.. 

\ 

Observed  declinations  arc  shou'n  by  dots. 

V 

Computed] 

{declinations 

by  first 

t jyeriodic , 

7 

icr 

rn'iula,by 

cu  rve. 

r 

1 

1 

1 

M 

J 

■ 

4 

r 

Fig,  7. 


old  determinations  of  the  declination  unless  the  rate  of  change 
be  known,  and  even  then  this  rate  is  not  likely  to  be  constant 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  23 


a great  many  years.  They  also  show  the  necessity  of  record- 
ing the  date  and  the  declination  of  the  needle  on  all  plats  and 
records  of  surveys,  with  a note  stating  whether  the  bearings 
given  were  the  true  or  magnetic  bearings  at  the  time  they 
were  taken. 

31.  Isogonic  Lines  are  imaginary  lines  on  the  earth’s  sur- 
face joining  points  whose  declinations  are  equal  at  any  given 
time.  The  isogonic  line  joining  points  having  no  declination 
is  called  the  agonic  line.  There  is  such  a line  crossing  the 
United  States  passing  just  east  of  Charleston,  S.  C.,  and  just 
west  of  Detroit,  Mich.  All  points  east  of  this  line  have  a 
western  declination,  and  all  points  west  of  it  have  an  eastern 
declination.  The  isogonic  lines  for  1885  for  the  whole  of 
the  United  States  are  shown  on  Plate  It  will  be  noted 

that  where  the  observations  are  most  thickly  distributed, 
as  in  Missouri  for  instance,  there  the  isogonic  lines  are  most 
crooked ; showing  that  if  the  declinations  were  accurately 
known  for  all  points  of  this  map  the  isogonic  lines  would  be 
much  more  irregular,  and  would  be  changed  very  much  in 
position  in  many  places. 

The  isogonic  lines  given  on  this  chart  are  all  moving  west- 
ward, so  that  all  western  declinations  are  increasing  and  all 
eastern  declinations  are  decreasing.  They  are  not  all  moving  at 
the  same  rate,  however,  those  in  New  Brunswick  and  also  those 
near  the  eastern  boundaries  of  California  and  Oregon  being 
about  stationary.  P'or  many  points  in  the  United  States  and 
Canada  the  rate  of  change  in  the  declination  has  been  observed, 
and  formulae  determined  for  computing  the  declination  for  each 
point,  which  formulae  will  probably  remain  good  for  the  next 
twenty  years.  The  following  tables  t give  this  information.  In 
these  tables  t is  the  time  in  calendar  years.  Thus  for  July  i, 
1885,  ^==1885.5.  In  the  first  table  all  the  formulae  have  been  re- 


* Reduced  from  charts  in  the  U.  S.  Coast  and  Geodetic  Survey  Report  for  1882. 
f Taken  from  the  above  report. 


24 


SURVEYING. 


ferred  to  one  date — Jan.  i,  1850.  Here  m is  used  to  represent 
the  time  in  years  after  1850,  or  ;;/=/— 1850.  Thus,  for  July  i, 
1885,  m = 35.5.  The  annual  value  of  this  secular  change  in  the 
declination  is  marked  at  various  points  on  the  isogonic  chart 
given  in  Plate  I.,  but  from  the  small  number  of  the  observa- 
tions, both  in  time  and  space,  it  is  evident  that  no  great  reli- 
ance can  be  placed  on  any  such  chart  for  exact  information. 

It  will  be  seen  that  the  change  in  the  declination  over  the 
Northern  States  will  average  about  one  minute  to  the  mile  in  an 
east  and  west  direction.  A value  of  the  declination  found  in 
one  end  of  a county  may  be  some  forty  minutes  in  error  in  the 
other  end  of  the  same  county.  This  shows  that  the  declina- 
tion must  be  known  for  the  exact  locality  of  the  survey.  In 
fact,  the  surveyor  can  never  be  sure  of  his  declination  until  he 
has  observed  it  for  himself  for  the  given  time  and  place.  This 
is  best  done  by  means  of  a transit  instrument,  and  such  a 
method  is  given  in  the  chapter  on  Geodetic  Surveying.  If, 
however,  no  transit  is  at  hand,  a result  sufficiently  accurate  for 
compass  surveying  may  be  obtained  by  the  compass  itself. 


FORMULA  EXPRESSING  THE  MAGNETIC  DECLINATION  AT  VARIOUS  PLACES. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  25 


FORMULA  EXPRESSING  THE  MAGNETIC  DECLINATION  AT  VARIOUS  PLACES- 

Continued. 


26 


SURVEYING. 


Approximate  expression. 


FORMULAE  EXPRESSING  THE  MAGNETIC  DECLINATION  AT  VARIOUS  PLACES— Continued. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  2/ 


* Approximate  expression. 


28 


SURVEYING. 


* There  are  pfiven  in  the  U.  S.  Coast  and  Geodetic  Survey  Report  for  1882,  and  issued  as  a separate  pamphlet,  the  declinations  of 
the  magnetic  needle  at  some  2500  points,  mostly  in  the  United  States,  all  reduced  to  the  epoch  Jan.  i,  1885. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  2g 


32.  Other  Variations  of  the  Declination. — In  addition 
to  the  daily  and  secular  changes  in  the  declination,  there  are 
others  worthy  of  mention. 

T/ie  annual  variation  is  very  small,  being  only  about  a half- 
minute of  arc  from  the  mean  position  for  the  year.  It  may 
therefore  be  neglected. 

The  lunar  ineqiialities  are  still  smaller,  being  only  about  fif- 
teen seconds  of  arc  from  the  mean  position. 

Magnetic  disturbances  are  due  to  what  are  called  magnetic 
storms.  They  may  occur  at  any  time,  and  cannot  be  predicted. 
They  may  last  a few  hours,  or  even  several  days.  “ The  fol- 
lowing table  of  the  observed  disturbances,  in  a bi-hourly  series, 
at  Philadelphia,  in  the  years  1840  to  1845,  will  give  an  idea  of 
their  relative  frequency  and  magnitude  : 


Deviations  from  nor- 
mal direction. 

Number  of 
disturbances. 

3'. 6 to  10'. 8 

2189 

10'.  8 to  18'.  I 

147 

18'.  I to  25'. 3 

18 

25'. 3 to  32'. 6 

3 

Beyond 

0 

At  Madison,  Wis.,  where  the  horizontal  magnetic  intensity 
is  considerably  less,  very  much  larger  deflections  have  been 
noticed.  Thus,  on  October  12,  1877,  4^^  ^^id  on  Ma^y 

28,  1877,  one  of  1°  24',  were  observed.”  * 

The  geometric  axis  of  a needle  may  not  coincide  with  its 
magnetic  axis,  and  hence  the  readings  of  two  instruments  at 
the  same  station  may  differ  slightly  when  both  are  in  adjust- 
ment. In  this  case  the  declination  should  be  found  for  each 
instrument  independently. 

33.  To  Find  the  Declination  of  the  Needle. — The 


- From  Report  of  the  U.  S.  Coast  and  Geodetic  Survey-for  1882. 


30 


SURVEYING, 


method  here  given  is  by  means  of  the  compass  and  a plumb- 
line,  and  is  sufficiently  accurate  for  compass-work.  The  com- 
pass-sights are  brought  into  line  with  the  plumb-line  and  the 
pole-star  (Polaris),  when  this  is  at  either  eastern  or  western 
elongation.  This  star  appears  to  revolve  in  an  orbit  of  i°  i8' 
radius.  Its  upper  and  lower  positions  are  called  its  upper  and 
lower  culminations,  and  its  extreme  east  and  west  positions  are 
called  its  eastern  and  western  elongations,  respectively.  When 
it  is  at  elongation  it  ceases  to  have  a lateral  component  of 
motion,  and  moves  vertically  upward  at  eastern  and  downward 
at  western  elongation.  If  the  star  be  observed  at  elongation, 
therefore,  the  observer’s  watch  may  be  as  much  as  ten  or 
fifteen  minutes  in  error,  without  its  making  any  appreciable 
error  in  the  result.  The  method  of  making  the  observation  is 
as  follows : 

Suspend  a fine  plumb-line,  such  as  an  ordinary  fishing-line, 
by  a heavy  weight  swinging  freely  in  a vessel  of  water.  The 
line  should  be  suspended  from  a rigid  point  some  fifteen  or 
twenty  feet  from  the  ground.  Care  must  be  taken  to  see  that 
the  line  does  not  stretch  so  as  to  allow  the  weight  to  touch  the 
bottom  of  the  vessel.  Just  south  of  this  line  set  two  stakes  in 
the  ground  in  an  east  and  west  direction,  leaving  their  tops  at 
an  elevation  of  four  or  five  feet.  Nail  to  these  stakes  a board 
on  which  the  compass  is  to  rest.  The  top  of  this  board  should 
be  smooth  and  level.  This  compass-support  should  be  as  far 
south  of  the  plumb-line  as  possible,  to  enable  the  pole-star  to 
be  seen  below  the  line-support.  A sort  of  wooden  box  may 
be  provided,  in  which  the  compass  is  rigidly  fitted  and  levelled. 
Several  hundred  feet  of  nearly  level  ground  should  be  open  to 
the  northward,  for  setting  the  azimuth-stake.  Prepare  two 
stakes,  tacks,  and  lanterns.  Find  from  the  table  given  on  page 
32  the  time  of  elongation  of  the  star.  About  twenty  minutes 
before  this  time,  set  the  compass  upon  the  board,  bringing  both 
sights  in  the  plane  defined  by  the  plumb-line  and  star.  The 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  3 1 


line  must  be  illuminated.  The  star  will  be  found  to  move 
slowly  east  or  west,  according  as  it  is  approaching  its  eastern 
or  western  elongation.  When  it  ceases  to  move  laterally,  the 
compass  is  carefully  levelled,  the  rear  compass-sight  brought 
into  the  plane  of  the  line  and  star,  and  then  the  forward  com- 
pass-sight made  to  coincide  with  the  rear  sight  and  plumb-line. 
(If  the  forward  sight  were  tall  enough,  we  could  at  once  bring 
both  slits  into  coincidence  with  line  and  star.)  Continue  to  ex- 
amine rear  sight,  line,  and  star,  and  rear  sight,  forward  sight, 
and  line  alternately,  until  all  are  found  to  be  in  perfect  coinci- 
dence, the  instrument  still  being  level.  If  this  is  completed 
within  fifteen  minutes  of  the  true  local  time  of  elongation,  the 
observation  may  be  considered  good ; and  if  it  is  completed 
within  thirty  minutes  of  the  time  of  elongation,  the  resulting 
error  in  azimuth  will  be  less  than  one  minute  of  arc.  Having 
completed  these  observations,  remove  the  plumb-line  and  set  a 
stake  in  the  line  of  sight  as  given  by  the  compass,  several  hun- 
dred feet  away.  In  the  top  of  this  stake  a tack  is  to  be  set 
exactly  on  line.  For  setting  this  tack,  a board  may  be  used, 
having  a vertical  slit  about  J inch  wide,  covered  with  white 
cloth  or  paper,  behind  which  a lamp  is  held.  This  slit  can 
then  be  accurately  aligned  and  the  tack  set.  A small  stake 
with  tack  is  now  set  just  under  the  compass  (or  plumb-line), 
and  the  work  is  complete  for  the  night.  Great  care  must  be 
taken  not  to  disturb  the  compass  after  its  final  setting  on  the  line 
and  star. 

At  about  ten  o’clock  on  the  following  day,  mount  the  com- 
pass over  the  south  stake.  From  the  north  stake  lay  off  a line 
at  right  angles  to  the  line  joining  the  two  stakes  (by  compass, 
optical  square,  or  otherwise)  towards  the  west  if  eastern 
elongation,  or  towards  the  east  if  western  elongation  had  been 
observed.  Carefully  measure  the  distance  between  the  two 
stakes  by  some  standardized  unit.  From  the  table  of  azimuths 
on  page  33  find  the  azimuth  of  the  star  at  elongation  for  the 


32 


SURVEYING. 


given  time  and  latitude.  Multiply  the  tangent  of  this  angle 
by  the  measured  distances  between  the  stakes,  and  care- 
fully lay  it  off  from  the  north  tack,  setting  a stake  and  tack. 
This  is  now  in  the  meridian  with  the  south  point.  With  the 
compass  in  good  adjustment,  especially  as  to  the  bubbles  and 
the  verticality  of  the  sights,  the  observation  for  declination 
may  now  be  made.  If  this  be  done  at  about  10.30  A.M.,  it 
will  give  the  mean  daily  declination.  Many  readings  should 
be  made,  allowing  the  needle  to  settle  independently  each  time. 
The  fractional  part  of  a division  on  the  graduated  limb  should 
be  read  by  the  declination-vernier,  thus  enabling  the  needle  to 
be  set  exactly  at  a graduation-mark.  If  all  parts  of  this  work 
be  well  done,  it  will  give  the  declination  as  accurately  as  the 
flag  can  be  set  by  means  of  the  open  sights. 

MEAN  LOCAL  TIME  (ASTRONOMICAL,  COUNTING  FROM  NOON) 
OF  THE  ELONGATIONS  OF  POLARIS. 


[The  table  answers  directly  for  the  year  1885,  and  for  latitude  -|-  40°.] 


Date. 

Eastern 

Elongation. 

Western 

Elongation. 

Date. 

Eastern 

Elongation. 

Western 

Elongation. 

Jan. 

I 

0'' 

35“-3 

12^ 

24” 

.6 

July 

I 

12^ 

39° 

^6 

32™.  8 

“ 

15 

23 

36  .1 

II 

29 

.3 

i ( 

15 

II 

44 

•7 

23 

34  -o 

Feb. 

I 

22 

29  .0 

10 

22 

.2 

Aug. 

I 

10 

38 

.2 

22 

27  .5 

“ 

15 

21 

33  .7 

9 

27 

.0 

“ 

15 

9 

43 

•3 

21 

32  .6 

Mar. 

I 

20 

38  .5 

8 

31 

.8 

Sept. 

I 

8 

36 

•7 

20 

26  .0 

i i 

15 

19 

43  -4 

7 

36 

.6 

i “ 

15 

7 

41 

.7 

19 

31  -I 

Apr. 

I 

18 

36  .4 

6 

29 

•7 

Oct. 

I 

6 

38 

•9 

18 

28  .2 

i ( 

15 

17 

41  .4 

5 

34 

•7 

1 “ 

15 

5 

43 

•9 

17 

33  .2 

May 

I 

16 

38  .6 

4 

31 

.8 

Nov. 

I 

4 

37 

.0 

16 

26  .4 

( < 

15 

15 

43  .7 

3 

36 

•9 

“ 

15 

3 

41 

•9 

15 

31  -3 

June 

I 

14 

37  -I 

2 

30 

•3 

Dec. 

I 

2 

38 

•9 

14 

28  .2 

* * 

15 

13 

42  .2 

I 

35 

•4 

15 

I 

43 

.6 

13 

33  -o 

ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  33 


AZIMUTH  (FROM  THE  NORTH)  OF  POLARIS,  WHEN  AT  ELONGA- 
TION, BETWEEN  THE  YEARS  1886  AND  1895,  FOR  DIFFERENT 
LATITUDES  BETWEEN  + 25°  AND  + 50". 


Lat. 

1886.0 

00 

00 

b 

i838.o 

1889.0 

1890.0 

1891.0 

1892.0 

1893.0 

1894.0 

1895.0 

0 

0 / 

c / 

0 / 

0 / 

0 / 

0 / 

0 / 

0 / 

0 / 

0 / 

+ 25 

I 26.0 

I 25.7 

I 25.3 

I 25.0 

I 24.6 

I 24.3 

I 23.9 

I 23.6 

I 23.2 

I 22.9 

26 

26.7 

26.4 

26.0 

25.7 

25-3 

25.0 

24.6 

24.3 

23.9 

23.6 

27 

27-5 

27.1 

26.8 

26.4 

26.0 

25-7 

25.4 

25.1 

24.7 

24-3 

28 

28.3 

27.9 

27.6 

27.2 

26.8 

26.5 

26.2 

25.8 

25.4 

25 -ij 

29 

29.1 

28.8 

28.4 

28.0 

27.6 

27-3 

27.0 

26.6 

26.3 

25.9 

30 

30.0 

29.6 

29-3 

28.9 

28.5 

28.2 

27.8 

27-5 

27.1 

26.8 

31 

30.9 

30.5 

30.2 

29.8 

29.4 

29. 1 

28.8 

28.4 

28.0 

27.6 

32 

31-9 

31-5 

31.2 

30.8 

30.4 

30.1 

29.7 

29-3 

29.0 

28.6 

33 

33-0 

32.6 

32.2 

31.8 

31.4 

31*1 

30.7 

30.3 

30.0 

29.6 

34 

34.0 

33-6 

33*3 

32.9 

32.5 

32.1 

31.8 

31-4 

31.0 

30.6 

35 

35.2 

34  8 

34-4 

34.0 

33.6 

33-2 

32.9 

32.5 

32.1 

31-7 

36 

36.4 

36.0 

35-6 

35-2 

34-8 

34-4 

34.0 

33.6 

33.2 

32.9 

37 

37-6 

37.2 

36.8 

36.4 

36.0 

35-6 

35-2 

34-8 

34.5 

34.1 

38 

38.9 

38.5 

00 

37.7 

37.3 

36.9 

36.5 

36.1 

35.7 

35.3 

39 

40.3 

39-9 

39-5 

39-1 

38.7 

38.3 

37-9 

37.5 

37.1 

36.7 

40 

41.8 

41.4 

41.0 

40.5 

40. 1 

39*7 

39-3 

38.9 

38.5 

38.1 

41 

43-3 

42.9 

42.5 

42.0 

41.6 

41.2 

40.8 

40.4 

40.0 

39-6 

42 

44.9 

44-5 

44.1 

43-6 

43-2 

42.8 

42.4 

42.0 

41.5 

41. 1 

43 

46.6 

46.1 

45*7 

45.3 

44.9 

44.4 

44.0 

43-6 

43.2 

42.7 

44 

48.4 

47-9 

47-5 

47.1 

46.6 

46.2 

45-8 

45-3 

44.9 

44.4 

45 

50.3 

49.8 

49.4 

48.9 

48.5 

48.1 

47.6 

47.1 

46.7 

46.2 

46 

52.2 

51.8 

51.3 

50.9 

50.4 

50.0 

49-5 

49.0 

48.6 

48.2 

47 

54.3 

53.8 

53-4 

52.9 

52.5 

52.0 

51.5 

51.0 

50.6 

50.2 

48 

56.5 

56.0 

55-6 

55.1 

54-6 

54.2 

53-7 

53-2 

52.8 

52.3 

49 

I 58.8 

I 58.3 

I 57.9 

57-4 

56.9 

56.5 

56.0 

55-5 

55-0 

54-5 

+ 50 

2 01.3 

2 00.8|2  00.3 

I 59.8 

I 59-3 

I 58.8 

I 58.4 

I 57.9 

I 57-4 

I 56.9 

3 


34 


SUR  VE  YING. 


If  the  elongation  of  Polaris  does  not  come  at  a suitable 
time  for  observing  for  declination,  the  upper  culmination,  which 
occurs  5*'  54™*^  after  the  eastern,  or  the  lower  culmination, 
& 03"\4  after  the  western  elongation,  may  be  chosen.  The 
objection  to  this  is  that  the  star  is  then  moving  at  its  most 
rapid  rate  in  azimuth.  It  is  so  near  the  pole,  however,  that  if 
the  observation  can  be  obtained  within  two  minutes  of  the 
time  of  its  culmination  the  resulting  error  will  be  less  than  T 
of  arc.  This  will  then  give  the  true  meridian  without  having 
to  make  offsets. 

It  must  be  remembered  that  the  time  of  elongation  given 
in  the  table  is  the  local  time  at  the  place  of  observation.  In- 
asmuch as  hourly  meridian  time  is  now  carried  at  most  points 
in  this  country  to  the  complete  exclusion  of  local  time,  it  will 
be  necessary  to  find  the  local  time  from  the  known  meridian  or 
watch  time.  Thus,  all  points  in  the  United  States  east  of  Pitt.s- 
burgh  use  the  fifth-hour  meridian  time  (75°  w.  of  Greenwich); 
from  Pittsburgh  to  Denver,  the  sixth-hour  meridian  time  (90° 
w.  of  Greenwich),  etc.  To  find  local  time,  therefore,  the  longi- 
tude east  or  west  of  the  given  meridian  must  be  found.  This 
can  be  determined  with  sufficient  accuracy  from  a map.  Thus, 
if  the  longitude  of  the  place  is  80°  w.  from  Greenwich,  it  is 
5°  w.  of  the  fifth-hour  meridian,  or  local  time  is  twenty  min- 
utes slower  than  meridian  time  at  that  place  If  meridian  time 
is  used  at  such  a place,  the  elongation  will  occur  twenty  min- 
utes later  than  given  by  the  table.  If  the  longitude  from 
Washington  is  given  on  the  map  consulted,  add  it  to  77°  if 
west  of  Washington,  and  subtract  it  from  77°  if  east  of  Wash- 
ington, to  get  longitude  from  Greenwich. 

USE  OF  THE  NEEDLE-COMPASS. 

34.  The  Use  of  the  Needle-compass  is  confined  almost 
exclusively  to  land-surveying,  where  an  error  of  one  in  three 
hundred  could  be  allowed.  As  the  land  enhances  in  value. 


T 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  35 


however,  there  is  an  increasing  demand  for  more  accurate 
means  of  determining  areas  than  the  compass  and  chain  af- 
ford. The  original  U.  S.  land-surveys  were  all  made  with  the 
needle,  or  with  the  solar,  compass  and  Gunter’s  chain.  Hence 
all  land  boundaries  in  this  country  have  their  directions  given 
in  compass-bearings,  and  their  lengths  in  chains  of  sixty-six 
feet  each. 

The  compass  is  used,  therefore, — 

1.  To  establish  a line  of  a given  bearing. 

2.  To  determine  the  bearing  of  an  established  line. 

3.  To  retrace  old  lines. 

If  the  true  bearing  is  to  be  used,  the  declination  of  the 
needle  from  the  meridian  must  be  determined  and  set  off  by 
the  vernier. 

If  the  magnetic  bearing  is  used,  the  declination  of  the 
needle  at  the  time  the  survey  was  made  should  be  recorded 
on  the  plat. 

If  old  lines  are  to  be  retraced,  the  declinations  at  the  times 
of  both  surveys  must  be  known. 

The  needle  should  be  read  to  the  nearest  five  minutes. 
This  requires  reading  to  sixths  of  the  half-degree  spaces,  but 
this  can  be  done  with  a little  practice. 

Always  lift  the  needle  from  the  pivot  before  moving  the  in- 
strument. 

If  the  needle  is  sluggish  in  its  movements  and  settles  quickly 
it  has  either  lost  its  magnetic  force  or  it  has  a blunt  pivot.  In 
either  case  it  is  likely  to  settle  considerably  out  of  its  true  posi- 
tion. The  longer  a needle  is  in  settling  the  more  accurate  will 
be  its  final  position.  It  can  be  quickly  brought  very  near  its 
true  position  by  checking  its  motion  by  means  of  the  lifting 
screw.  In  its  final  settlement,  however,  it  must  be  left  free. 

Careful  attention  to  the  instrumental  adjustments,  to  local 
disturbances,  and  close  reading  of  the  needle  are  all  essential 
to  good  results  with  the  compass. 


36 


SURVEYING. 


< 


35*  "I  o set  off  the  Declination,  we  liave  only  to  remem- 
ber that  the  declination  arc  is  attached  to  the  line  of  sight  and 
that  the  vernier  is  attached  to  the  graduated  circle.  If  the 
declination  is  west,  then  when  the  line  of  sight  is  north  the 
north  end  of  the  needle  points  to  the  left  of  the  zero  of  the 
graduated  circle.  In  order  that  it  may  read  zero,  or  north,  the 
circle  must  be  moved  towards  the  left,  or  opposite  to  the  hands 
of  a watch.  On  the  other  hand,  if  the  declination  is  east,  the 
circle  to  which  the  vernier  is  attached  should  be  moved  with 
the  hands  of  a watch.  This  at  once  enables  the  observer  to 
set  the  vernier  so  that  the  needle-readings  will  be  the  true 
bearings  of  the  line  of  sight. 

36.  Local  Attractions  may  disturb  the  needle  by  large  or 
small  amounts,  and  these  often  come  from  unknown  causes. 
The  observer  should  have  them  constantly  in  mind,  and  keep  all 
iron  bodies  at  a distance  from  the  instrument  when  the  needle 
is  being  read.  The  glass  cover  may  become  electrified  from 
friction,  and  attract  the  needle.  This  can  be  discharged  by 
touching  it  with  a wet  finger,  or  by  breathing  upon  it.  Read- 
ing-glasses should  not  have  gutta-percha  frames,  as  these  be- 
come highly  electrified  by  wiping  the  lens,  and  will  attract  the 
needle.  Such  glasses  should  have  brass  or  German-silver 
frames.  No  nickel  coverings  or  ornaments  should  be  near,  as 
this  metal  has  magnetic  properties.  A steel  band  in  a hat- 
brim,  or  buttons  containing  iron,  have  been  known  to  cause 
great  disturbance.  In  cities  and  towns  it  is  practically  impos- 
sible to  get  away  from  the  influence  of  some  local  attraction, 
such  as  iron  or  gas  pipes  in  the  ground,  iron  lamp-posts,  fences, 
building-fronts,  etc.  For  this  reason  the  needle  should  never 
be  used  in  such  places. 

In  many  regions,  also,  there  are  large  magnetic  iron-ore  de- 
posits in  the  ground,  which  give  special  values  for  the  declina- 
tion at  each  consecutive  station  occupied.  It  is  practically 
impossible  to  use  magnetic  bearings  in  such  localities. 

77/^’  test  for  local  attraction  in  the  field-work  is  to  read  the 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS,  3/ 


bearing  of  every  line  from  both  ends  of  it.  If  these  are  not 
the  same,  and  no  error  has  been  made,  there  is  some  local  dis- 
turbance at  one  station  not  found  at  the  other.  If  there  is 
known  to  be  mineral  deposits  in  the  region  it  may  perhaps  be 
laid  to  that.  If  not,  the  preceding  station  should  be  occupied 
again,  and  the  cause  of  the  discrepancy  inquired  into.  If  the 
forward  and  reverse  bearings  of  all  lines  agree  except  the  bear- 
ings taken  from  a single  station,  then  it  may  be  assumed  there 
is  local  attraction  at  that  station. 

ELIMINATION  OF  LOCAL  ATTRACTIONS. 

37.  To  establish  a Line  of  a Given  Bearing,  set  the  com- 
pass up  at  a point  on  tiie  line,  turn  off  the  declination  on  the 
declination-arc,  and  bring  the  north  end  of  the  needle  to  the 
given  bearing.  The  line  of  sight  now  coincides  with  the  re- 
quired line,  and  other  points  can  be  set. 

38.  To  find  the  True  Bearing  of  a Line,  set  the  compass 
up  on  the  line,  turn  off  the  declination  by  the  vernier,  bring 
the  line  of  sight  to  coincide  with  the  line  with  the  south  part  of 
the  graduated  circle  towards  the  observer,  and  read  the  north 
end  of  the  needle.  This  gives  the  forward  bearing  of  the  line. 

39.  To  retrace  an  Old  Line,  set  the  compass  over  one 
well-determined  point  in  the  line  and  turn  the  line  of  sight 
upon  another  such  point.  Read  the  north  end  of  the  needle. 
If  this  reading  is  not  the  bearing  as  given  for  the  line,  move 
the  vernier  until  the  north  end  of  the  needle  comes  to  the 
given  bearing,  when  the  sights  are  on  line.  The  reading  of 
the  declination-arc  will  now  give. the  declination  to  be  used  in 
retracing  all  the  other  lines  of  the  same  survey.  If  a second 
well-determined  point  cannot  be  seen  from  the  instrument-sta- 
tion, a trial-line  will  have  to  be  run  on  an  assumed  value  for 
the  declination,  and  then  the  value  of  the  declination  used  on 
the  first  survey  computed.  Thus,  if  the  trial-line,  of  length  /, 
comes  out  a distance  x to  the  right  of  the  known  point  on 


38 


SURVEYING. 


the  line,  the  vernier  is  to  be  moved  in  the  direction  of  the  hands 

of  a watch  an  angular  amount  whose  tangent  is  j.  If  the 

trial-line  comes  out  to  the  /eyt  of  the  point,  move  the  vernier 
in  a direction  opposite  to  the  hands  of  a watch. 

PRISMATIC  COMPASS. 

40.  The  Prismatic  Compass  is  a hand-instrument  pro- 
vided with  a glass  prism  so  adjusted  that  the  needle  can  be 
read  while  taking  the  sight.  A convenient  form  is  shown  in 
Fig.  9,  which  is  carried  in  the  pocket  as  a watch.  The  line  of 


sight  is  established  by  means  of  the  etched  line  on  the  glass 
cover  5.  It  is  used  in  preliminary  and  reconnoissance  work,  in 
clearing  out  lines,  etc. 

EXERCISES  FOR  COMPASS  ALONE  OR  FOR  COMPASS  AND  CHAIN. 

41.  Run  out  a line  of  about  a mile  in  length,  on  somewhat  uneven  ground, 
establishing  several  stations  upon  it,  using  a constant  compass-bearing.  Then 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  39 


run  back  by  the  reve7'se  bearings  and  note  how  nearly  the  points  coincide  with  the 
former  ones.  The  chain  need  not  be  used. 

42.  Select  some  half  dozen  points  that  enclose  an  area  of  about  forty  acres 
(one  quarter  mile  square)  on  uneven  ground.  Let  one  party  make  a compass- 
and-chain  survey  of  it,  obtaining  bearing  and  length  of  each  side.  Then  let 
other  parties  take  these  field-notes  and,  all  starting  from  a common  point,  run 
out  the  lines  as  given  by  the  Jield-note'i,  setting  other  stakes  at  all  the  remaining 
corners,  each  party  leaving  special  marks  on  their  own  stakes.  Let  each  party 
plot  their  own  survey  and  compare  errors  of  closure. 

43.  Select  five  points,  three  of  which  are  free  from  local  attraction,  while  two 
consecutive  ones  are  known  to  be  subject  to  such  disturbance.  Make  the  sur- 
vey, finding  length  and  forward  and  reverse  bearings  of  every  side.  Determine 
what  the  true  bearing  of  each  course  is,  and  plot  to  obtain  the  error  of  closure. 

44.  Let  a number  of  parties  observe  for  the  declination  of  the  needle,  using 
a common  point  of  support  for  the  plumb-line.  Let  each  party  set  an  inde- 
pendent meridian  stake  in  line  with  the  common  point.  Note  the  distance  of 
each  stake  fro7n  the  77tea7t  positio7i,  and  compute  the  corresponding  angular  dis- 
crepancies. (March  and  September  are  favorable  months  for  making  these 
observations,  for  then  Polaris  comes  to  elongation  in  the  early  evening.) 

The  above  problems  are  intended  to  impress  upon  the  student  the  relative 
errors  to  which  his  work  is  subject. 

THE  SOLAR  COMPASS. 

45.  The  Burt  Solar  Compass  essentially  consists  first,  of 
a polar  axis  rigidly  attached  in  the  same  vertical  plane  with  a 
terrestrial  line  of  sight,  the  whole  turning  about  a vertical  axis. 
When  this  plane  coincides  with  the  meridian  plane,  the  polar 
axis  is  parallel  with  the  axis  of  the  earth.  Second,  attached 
to  the  polar  axis,  and  revolving  about  it,  is  a line  of  collimation 
making  an  angle  with  the  polar  axis  equal  to  the  angular  dis- 
tance of  the  sun  for  the  given  day  and  hour  from  the  pole. 
This  latter  angle  is  90°  plus  or  minus  the  sun’s  declination, 
according  as  the  sun  is  south  or  north  of  the  equator.  The 
polar  axis  must  therefore  make  an  angle  with  the  horizon 
equal  to  the  latitude  of  the  place,  and  the  line  of  collimation 
must  deviate  from  a perpendicular  to  this  axis  by  an  angular 
amount  equal  to,  and  in  the  direction  of,  the  sun’s  declination. 
With  these  angles  properly  set,  and  the  line  of  collimation 


40 


SUR  VE  YING. 


turned  upon  the  sun,  the  vertical  plane  through  the  terrestrial 
line  of  sight,  and  the  polar  axis  must  lie  in  the  meridian,  for 


otherwise  any  motion  of  the  line  of  collimation  about  its  axis 
would  not  bring  it  upon  the  sun. 

In  Fig.  lo  is  shown  a cut  of  this  instrument  as  manufac- 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  4 1 


tured  by  Young  & Sons,  Philadelphia.  The  polar  axis  is  shown 
at  /,  and  the  terrestrial  line  of  sight  is  defined  by  the  slits  in 
the  vertical  sights,  the  same  as  in  the  needle-compass.  The 
line  of  collimation  is  defined  by  a lens  at  the  upper  end  of  the 
arm  and  a silver  plate  at  the  lower  end,  containing  gradua- 
tions with  which  the  image  of  the  sun,  as  formed  by  the  lens, 
is  made  to  coincide.  The  polar  axis  is  given  the  proper  incli- 
nation by  means  of  the  latitude-arc  /,  and  the  line  of  collima- 
tion is  inclined  from  a perpendicular  to  this  axis  by  an  amount 
equal  to  the  sun’s  declination  by  means  of  the  declination-arc 
d.  When  these  arcs  are  properly  set,  the  arm  a is  revolved 
about  the  polar  axis,  and  the  whole  instrument  about  its  verti- 
cal axis,  until  the  image  of  the  sun  is  properly  fixed  on  the 
lines  of  the  silver  plate,  when  the  terrestrial  line  of  sight,  as 
defined  by  the  vertical  slits,  lies  in  the  true  meridian.  Any 
desired  bearing  may  now  be  turned  off  by  means  of  the  hori- 
zontal circle  and  vernier,  shown  at  v.  The  accuracy  with 
which  the  meridian  is  obtained  with  this  instrument  depends 
on  the  time  of  day,  and  on  the  accuracy  with  which  the  lati- 
tude- and  declination-angles  are  set  off.  It  is  necessary  to  at- 
tend carefully,  therefore,  to  the 

ADJUSTMENTS  OF  THE  SOLAR  COMPASS. 

46.  To  make  the  Plane  of  the  Bubbles  perpendicular  to 
the  Vertical  Axis. — This  is  done  by  reversals  about  the  verti- 
cal axis,  the  same  as  with  the  needle-compass. 

47.  To  adjust  the  Lines  of  Collimation. — The  declination- 
arm  a has  two  lines  of  collimation  that  should  be  made  paral- 
lel. As  it  is  shown  in  the  figure,  it  is  set  for  a south  declina- 
tion. This  is  the  position  it  will  occupy  from  Sept.  20  to 
March  20.  When  the  sun  has  a north  declination,  as  from 
March  20  to  Sept.  20,  the  declination-arm  is  revolved  180° 
about  the  polar  axis,  and  a line  of  collimation  established  by 


42 


SUR  VE  Y/NG. 


a lens  and  a graduated  disk  on  opposite  ends  from  those  pre- 
viously used.  Each  end  of  this  arm,  therefore,  has  both  a lens 
and  a disk,  each  set  of  which  establishes  a line  of  collimation. 
The  second  adjustment  consists  in  making  these  tivo  lines  of  col- 
limation parallel  to  each  other.  They  are  made  parallel  to  each 
other  by  making  both  parallel  to  the  faces  of  the  blocks  con- 
taining the  lenses  and  disks.  To  effect  this,  the  arm  must  be 
detached  and  laid  upon  an  auxiliary  frame  which  is  attached 
in  the  place  of  the  arm,  and  which  is  called  an  adjuster.  With 
the  latitude-  and  declination-arc  set  approximately  for  the  given 
time  and  place,  lay  the  declination-arm  upon  the  adjuster,  and 
bring  the  sun’s  image  upon  the  disk.  Now  turn  the  arm  care- 
fully bottom  side  up  (not  end  for  end)  and  see  if  the  sun’s 
image  comes  between  the  equatorial  lines  on  the  disk.*  If  not, 
adjust  the  disk  for  one  half  the  displacement,  and  reverse  again 
for  a check.  When  this  disk  is  adjusted,  turn  the  arm  end  for 
end,  and  adjust  the  other  disk  in  a similar  manner.  Having 
now  made  both  lines  of  collimation  parallel  to  the  edges  of  the 
blocks,  they  are  parallel  to  each  other. 

48.  To  make  the  Declination-arc  read  Zero  when  the 
Line  of  Collimation  is  at  Right  Angles  to  the  Polar  Axis. 
— Set  the  vernier  on  the  declination-arc  to  read  zero.  By  any 
means  bring  the  line  of  collimation  upon  the  sun.  When 
carefully  centred  on  the  disk,  revolve  the  arm  180°  quickly 
about  the  polar  axis,  and  see  if  the  image  now  falls  exactly 
on  the  other  disk.  If  not,  move  the  declination-arm  by 
means  of  the  tangent-screw  until  the  image  falls  exactly  on 
the  disk.  Read  the  declination-arc,  loosen  the  screws  in  the 
vernier-plate,  and  move  it  back  over  one  half  its  distance 
from  the  zero-reading.  Centre  the  image  again,  reverse  180°, 
and  test.  This  adjustment  depends  on  the  parallelism  of  the 
two  lines  of  collimation.  If  the  vernier-scale  is  not  adjustable, 

* It  would  not  be  expected  to  fall  between  the  hour-lines  on  the  disk,  since 
some  time  has  elapsed. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  43 


one  half  the  total  movement  is  the  index  error  of  the  declina- 
tion-arc, and  must  be  taken  into  account  in  all  settings  on  this 
arc. 

The  two  preceding  adjustments  should  be  made  near  the 
middle  of  the  day. 

49.  To  adjust  the  Vernier  of  the  Latitude-arc. — Find  the 
latitude  of  the  place,  either  from  a good  map  or  by  a transit- 
observation.  Set  up  the  compass  a few  minutes  before  noon, 
with  the  true  declination  set  off  for  the  given  day  and  hour. 
Bring  the  line  of  collimation  upon  the  sun,  having  it  clamped 
in  the  plane  of  the  sights,  or  at  the  twelve-hour  angle,  and 
follow  it  by  moving  the  latitude-arc  by  means  of  the  tangent- 
screw,  and  by  turning  the  instrument  on  its  vertical  axis. 
When  the  sun  has  attained  its  highest  altitude,  read  the  lati- 
tude-arc? Compare  this  with  the  known  latitude.  Move  the 
vernier  on  this  arc  until  it  reads  the  true  latitude ; or,  if  this 
cannot  be  done,  the  difference  is  the  index  error  of  the  latitude- 
arc.  If,  however,  the  latitude  used  with  the  instrument  be 
that  obtained  by  it,  as  above  described,  then  no  attention  need 
be  paid  to  this  error.  This  error  is  only  important  when  the 
true  latitude  is  used  with  the  instrument  in  finding  the  meridian, 
or  where  the  true  latitude  of  the  place  is  to  be  found  by  the  in- 
strument. In  using  the  solar  compass,  therefore,  ahvays  use 
the  latitude  as  given  by  that  instrument  by  a meridian  observa- 
tion on  the  sun."^ 

50.  To  make  the  Terrestrial  Line  of  Sight  and  the  Polar 
Axis  lie  in  the  same  Vertical  Plane. — This  should  be  done  by 
the  maker.  The  vertical  plane  that  is  really  brought  into  the 
meridian  by  a solar  observation  is  that  containing  the  polar 
axis,  and  by  as  much  as  tJie  plane  of  the  sights  deviates  from 


* Since  the  sun  may  cross  the  meridian  as  much  as  15  minutes  or  more 
before  or  after  mean  noon,  this  observation  may  have  to  be  taken  that  much 
before  or  after  12  o’clock  mean  time.  It  is,  however,  in  all  cases,  an  observation 
on  the  sun  ai  culmination. 


44 


SUR  VE  YING. 


this  plane,  by  so  much  will  all  bearings  be  in  error.  TIic  best 
test  of  this  adjustment  is  to  establish  a true  meridian  by  the 
transit  by  observations  on  a circumpolar  star  ; and  then  by 
making  many  observations  on  this  line,  in  both  forenoon  and 
afternoon,  one  may  determine  whether  or  not  the  horizontal 
bearings  should  have  an  index-correction  applied. 

USE  OF  THE  SOLAR  COMPASS. 

51.  The  Solar  Compass  is  used  on  land  and  other  surveys 
where  the  needle-compass  is  either  too  inaccurate,  or  where, 
from  local  attraction,  the  declination  of  the  needle  is  too  vari- 
able to  be  accurately  determined  for  all  points  in  the  survey. 
Where  there  is  no  local  attraction,  however,  and  the  declination 
of  the  needle  is  well  known,  the  advantages  of  the  solar  com- 
pass in  accuracy  are  fairly  offset  by  several  disadvantages  in  its 
use  which  do  not  obtain  with  the  needle-compass.  Thus,  the 
solar  compass  should  never  be  used  when  the  sun  is  less  than 
one  hour  above  the  horizon,  or  less  than  one  hour  from  noon. 
Of  course  it  cannot  be  used  in  cloudy  weather.  For  such  times 
as  these  bearings  may  be  obtained  by  a needle  which  is  always 
attached,  but  then  the  instrument  becomes  a needle-com- 
pass simply.  It  is  also  much  more  trouble,  and  consumes 
more  time  in  the  field  than  the  needle-compass.  But  more 
significant  than  any  of  these  is  the  fact  that  if  the  adjustments 
are  not  carefully  attended  to,  the  error  in  the  bearing  of  a line 
may  be  much  greater  by  the  solar  compass  than  is  likely  to 
be  made  by  the  needle-compass,  when  there  is  no  local  attrac- 
tion. It  is  possible,  however,  to  do  much  better  work  with 
the  solar  compass  than  can  be  done  with  the  needle-com- 
pass. 

52.  To  find  the  Declination  of  the  Sun. — On  account  of 
the  inclination  of  the  earth’s  axis  to  the  plane  of  its  orbit,  the 
sun  is  seen  north  of  the  celestial  equator  in  summer,  and  south 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  45 


of  it  in  winter.  This  deviation,  north  or  south  of  the  equator, 
is  called  north  or  south  declination,  and  is  measured  from  any 
point  on  the  earth’s  surface  in  degrees  of  arc. 

On  about  the  2ist  of  June  the  sun  reaches  its  most  northern 
declination,  and  begins  slowly  to  return.  Its  most  southern 
point  is  reached  about  December  2ist.  In  June  and  Decem- 
ber, therefore,  the  sun  is  changing  its  declination  most  slowly, 
while  at  the  intervening  quadrant-points  of  the  earth’s  orbit, 
March  and  September,  it  is  changing  its  declination  most 
rapidly,  being  as  much  as  one  minute  in  arc  for  one  hour  in 
time.  It  is  evident,  therefore,  that  we  must  regard  the  decli- 
nation of  the  sun  as  a constantly  changing  quantity,  and, 
for  any  given  day’s  work,  a table  of  declinations  must  be 
made  out  for  each  hour  of  the  day.  The  American  Ephemeris 
and  Nautical  Almanac  gives  the  declination  of  the  sun  for  noon 
of  each  day  of  the  year  for  both  Greenwich  and  Washington. 
Since  the  time  universally  used  in  this  country  is  so  many 
hours  from  Greenwich,  it  is  best  to  use  the  Greenwich  declina- 
tions. Since,  also,  we  are  five,  six,  seven,  or  eight  hours  west 
of  Greenwich,  the  declination  given  in  the  almanac  for  Green- 
wich noon  of  any  day  will  correspond  to  the  declination  here 

7,  6,  5,  or  4 o’clock  A.M.  of  the  same  date,  according  as  East- 
ern, Central,  Mountain,  or  Western  time  is  used.  As  this 
standard  time  is  seldom  more  than  30  minutes  different  from 
local  time,  and  as  this  could  never  affect  the  declination  by  more 
than  30  seconds  of  arc,  it  will  here  be  considered  sufficient  to 
correct  the  Greenwich  declination  by  the  change,  as  found  for 
the  standard  time  used.  Thus,  if  Central  (90th  meridian)  time 
is  used,  the  declination  given  in  the  almanac  is  the  declination 
at  6 o’clock  A.M.  at  the  place  of  observation.  To  this  must  be 
added  (algebraically)  the  hourly  change  in  declination,  which  is 
also  given  in  the  almanac.  A table  may  thus  be  prepared,  giv- 
ing the  declination  for  each  hour  of  the  day. 

53.  T o correct  the  Declination  for  Refraction. — All  rays 


46 


SURVEYING. 


of  light  coming  to  the  earth  from  exterior  bodies  are  refracted 
downward,  thus  causing  such  bodies  to  appear  higher  than 
they  really  are.  This  refraction  is  zero  for  normal  (vertical) 
lines,  and  increases  towards  the  horizon.  It  varies  largely, 
also,  with  the  special  temperature,  pressure,  and  hydrometrical 
condition  of  the  atmosphere.  Tables  of  refraction  give  only 
the  mean  values,  and  these  may  differ  largely  from  the  values 
found  for  any  given  time,  especially  for  lines  near  the  horizon. 
It  is  for  this  reason  that  all  astronomical  observations  made 
near  the  horizon  are  very  uncertain.  There  is  but  one  setting 
on  the  solar  compass  that  has  reference  to  the  position  of  the 
sun  in  the  heavens,  and  that  is  the  declination.  Now,  the  re- 
fraction changes  the  apparent  altitude  of  the  body  ; and  by  so 
much  as  a change  in  the  altitude  changes  the  declination,  by 
so  much  does  the  apparent  declination  differ  from  the  true  dec- 
lination. Evidently  it  is  the  apparent  declination  that  should 
be  set  off.  When  the  sun  is  on  the  meridian,  the  change  in 
altitude  has  its  full  effect  in  changing  the  declination,  but  at 
other  times  the  change  in  declination  is  less  than  the  change 
in  altitude. 

The  correction  to  the  declination  due  to  refraction  is  found 
from  the  following  final  equations  : * 

tan  N — cot  cp  cos 


tan  q = 


sin  N 

cos  (d  N) 


tan  z 


cot  (d  + JV) 
cos  q ‘ 


dS  = ■—  dz  cos 


* See  Chauvenet’s  “ Spherical  Astronomy,”  vol.  i.,  p.  171,  and  Doolittle’s 
“ Practical  Astronomy,”  p.  159. 


ADJUSTMENT,  USE,  AND  CATE  OF  INSTRUMENTS.  4/ 


where  q)  = latitude ; t = hour  angle  from  the  meridian ; S = 
declination  of  sun  ; ^ — zenith  distance  of  sun;  TV  and  q being 
auxiliary  angles  to  facilitate  the  computation. 

From  these  equations  we  may  compute  the  auxiliary  angle 
q,  and  the  zenith  distance  z,  for  each  hour  from  noon,  for  every 
day  of  the  year.  Then  from  a table  of  mean  refractions,  giving 
the  refraction  for  given  altitudes,  or  zenith  distances,  which  is 
dz,  we  may  find  the  corresponding  d8,  which  is  the  correction 
to  be  applied  to  the  declination  for  refraction. 

In  this  manner  the  following  table  has  been  computed  for 
the  latitude  of  40°.  For  any  other  latitude  the  correction  is 
found  by  multiplying  the  correction  given  in  the  table  by  the 
corresponding  coefficient,  as  given  in  the  table  “ Latitude  Co- 
efficients.” These  coefficients  were  obtained  by  plotting  the 
ratios  of  the  actual  refraction  at  the  different  latitudes  to  that 
at  latitude  40°,  for  each  hour  from  7 A.M.  to  5 P.M.  and  for  the 
various  declinations.  It  was  found  that  this  ratio  was  almost 
a constant,  except  for  very  low  altitudes,  where  the  inherent 
uncertainties  of  an  observation  are  very  large,  from  the  actual 
refraction  varying  so  largely  from  the  mean,  as  given  in  the 
tables.  A mean  value  of  this  ratio  was  chosen,  therefore, 
which  enables  the  corrections  at  other  latitudes  to  be  found  in 
terms  of  those  in  latitude  40°  without  material  error.  These 
ratios  are  given  in  the  Table  of  Latitude  Coefficients. 

EXAMPLE. 

Let  it  be  required  to  prepare  a table  of  declination  settings 
for  a point  whose  latitude  is  38°  30',  which  lies  in  the  “ Central 
Time  Belt,”  and  for  April  5,  1890. 

Since  the  time  is  6 h.  earlier  than  that  at  Greenwich,  the 
declination  given  in  the  Ephemeris  for  Greenwich  mean  noon 
(6°  9'  57")  is  the  declination  for  the  given  place  at  6 A.M.  If 
the  point  were  in  the  “ Eastern  Time  Belt  ” it  would  be  the 
declination  at  7 A.M.,  etc.  Suppose  it  is  desired  to  prepare 
declination  settings  from  7 A.M.  to  5 P.M.  From  the  table  of 


48 


SUR  VE  YING. 


TABLE  OF  REFRACTION  CORRECTION  TO  BE  APPLIED  TO  THE 

DECLINATIONS. 


Refraction 

Refraction 

1 

Refraction 

Refraction 

Date. 

Correction. 

Date. 

Correction. 

Date. 

Correction. 

Date. 

Correction. 

Latitude  40°. 

Latitude  40°. 

Latitude  40“. 

Latitude  40“. 

Jan. 

Feb. 

Mar. 

May. 

*i  h. 

l' 

S8" 

13 

I h. 

i' 

16" 

30 

h. 

42" 

»4 

I 

h. 

23" 

2 

2 

16 

14 

2 

25 

31 

15 

2 

27 

3 

3 

04 

15 

3 

I 

48 

April. 

47 

57 

18 

16 

3 

34 

3 

4 

6 

23 

16 

17 

4 

5 

2 

8 

47 

39 

2 

3 

4 

5 

i* 

2 

*7 

i3 

4 

5 

i' 

U 

4 

5 

2 

2 

54 

II 

18 

19 

2 

I 

I 

12 

20 

4 

C 

1 

2 

39 

44 

54 

14 

08 

»9 

20 

1 

2 

22 

26 

3 

2 

59 

20 

3 

I 

40 

0 

6 

3 

21 

3 

33 

1 

4 

6 

01 

21 

4 

2 

31 

7 

A 

j 

22 

4 

47 

22 

5 

6 

49 

8 

5 

2 

23 

5 

I 

15 

9 

I 

51 

23 

T 

07 

9 

I 

36 

24 

I 

21 

2 

2 

07 

24 

2 

I 

15 

10 

2 

41 

25 

2 

25 

3 

2 

51 

25 

3 

1 

33 

II 

3 

51 

26 

3 

32 

12 

4 

5 

40 

26 

4 

2 

18 

12 

4 

I 

10 

27 

4 

46 

13 

27 

5 

5 

29 

13 

5 

58 

28 

5 

I 

13 

14 

15 

16 

17 

18 

1 

2 

3 

4 

1 

2 

2 

5 

46 

01 

40 

00 

28 

Mar. 

2 

3 

4 

1 

2 

3 

4 

5 

I 

1 

1 

2 

4 

03 

10 

27 

06 

39 

14 

ll 

*7 

18 

1 

2 

3 

4 

5 

I 

1 

34 

38 

48 

06 

49 

29 

30 

June. 

1 

2 

1 

2 

3 

4 

5 

I 

20 

24 

31 

44 

II 

19 

20 

I 

I 

42 

56 

5 

I 

0 

59 

19 

20 

1 

2 

32 

36 

3 

I 

19 

21 

2 

I 

6 

2 

I 

06 

21 

3 

45 

4 

2 

23 

22 

3 

2 

31 

7 

3 

I 

21 

22 

4 

I 

02 

5 

3 

30 

23 

4 

4 

35 

8 

4 

I 

56 

23 

5 

I 

42 

6 

4 

43 

9 

5 

4 

04 

7 

5 

I 

10 

24 

37 

24 

I 

30 

25 

I 

I 

10 

I 

55 

25 

2 

34 

8 

I 

t8 

26 

50 

II 

2 

I 

02 

26 

3 

42 

9 

2 

22 

27 

3 

2 

22 

12 

3 

I 

15 

27 

4 

58 

10 

3 

29 

28 

4 

4 

07 

13 

4 

I 

47 

28 

5 

I 

36 

II 

4 

43 

14 

5 

3 

34 

12 

5 

I 

09 

29 

29 

j 

28 

32 

39 

18 

30 

I 

I 

32 

15 

I 

52 

2 

13 

I 

2 

I 

44 

16 

2 

58 

May. 

■a 

14 

2 

22 

Feb. 

3 

2 

13 

17 

3 

I 

10 

I 

IS 

3 

29 

1 

2 

4 

3 

41 

18 

19 

4 

5 

1 

3 

39 

08 

2 

3 

4 

5 

I 

55 

30 

16 

17 

4 

5 

I 

42 

08 

3 

4 

I 

T 

26 

20 

21 

2 

48 

54 

4 

5 

1 

2 

26 

30 

18 

19 

1 

2 

18 

22 

5 

2 

37 

22 

3 

I 

OS 

6 

3 

37 

20 

3 

29 

6 

3 

04 

23 

4 

I 

32 

7 

4 

53 

21 

4 

42 

7 

4 

3 

21 

24 

5 

2 

51 

8 

5 

26 

22 

5 

08 

8 

I 

I 

21 

25 

I 

45 

9 

I 

25 

23 

I 

18 

9 

2 

I 

3* 

26 

2 

50 

10 

2 

29 

24 

2 

22 

10 

3 

1 

56 

27 

3 

I 

01 

II 

3 

36 

25 

3 

29 

11 

4 

3 

04 

28 

4 

I 

25 

12 

4 

51 

26 

4 

42 

12 

29 

5 

2 

34 

13 

5 

I 

22 

27 

5 

08 

* The  hours  are  counted  each  way  from  noon.  Thus  9 a.m.  and  3 p.m.  would  correspond  to 


the  3d  hour  in  the  table. 


SURVEYING. 


48^ 


Refraction 

Refraction 

Refraction 

Date. 

Correction. 

Date. 

Correction. 

Date. 

Correction. 

Date. 

Latitude  40®. 

Latitude  40®. 

Latitude  40®. 

June. 

Aug. 

Oct. 

Nov. 

28 

I h. 

18" 

17 

I h. 

32" 

6 

I 

h.  1' 

03" 

20 

29 

18 

2 

36 

7 

2 

I 

10 

21 

jSy. 

3 

29 

43 

09 

19 

20 

3 

4 

1^ 

. 45 

02 

8 

9 

3 

4 

1 

2 

27 

06 

22 

23 

1 

2 

5 

i' 

21 

5 

I 

42 

10 

5 

4 

39 

24 

22 

I 

34 

3 

I 

19 

23 

2 

38 

II 

I 

I 

07 

25 

26 

4 

2 

23 

24 

3 

48 

12 

2 

I 

15 

5 

3 

30 

25 

4 

I 

06 

13 

3 

I 

33 

27 

28 

6 

4 

43 

26 

5 

I 

49 

14 

4 

2 

18 

7 

5 

I 

10 

27 

I 

36 

15 

5 

5 

29 

29 

8 

I 

20 

28 

2 

41 

9 

2 

24 

29 

3 

51 

16 

I 

I 

12 

10 

3 

31 

30 

4 

I 

10 

17 

2 

1 

20 

II 

4 

44 

31 

5 

I 

58 

18 

3 

I 

40 

Dec. 

12 

5 

I 

II 

19 

4 

2 

31 

I 

Sept. 

20 

5 

6 

49 

2 

»3 

I 

21 

I 

I 

39 

3 

14 

2 

25 

2 

2 

44 

16 

4 

15 

3 

32 

3 

3 

54 

21 

I 

I 

16 

4 

46 

4 

4 

I 

14 

22 

2 

I 

25 

17 

5 

I 

13 

5 

5 

2 

08 

23 

3 

I 

48 

e 

18 

24 

4 

2 

47 

6 

I 

22 

6 

I 

42 

25 

5 

8 

39 

19 

2 

26 

7 

2 

47 

7 

8 

20 

3 

33 

8 

3 

57 

2l 

4 

47 

9 

4 

I 

19 

26 

I 

I 

21 

9 

22 

5 

I 

15 

10 

5 

2 

18 

27 

2 

I 

31 

48 

28 

3 

I 

56 

23 

I 

23 

II 

I 

29 

4 

3 

04 

10 

24 

2 

27 

12 

2 

50 

30 

5 

II 

01 

II 

25 

3 

34 

13 

3 

I 

01 

12 

26 

4 

14 

4 

I 

25 

N 

26 

13 

27 

5 

I 

18 

15 

5 

2 

34 

Nov. 

2 

I 

37 

14 

28 

25 

29 

36 

51 

22 

16 

I 

48 

2 

3 

2 

04 

29 

30 

1 

2 

3 

17 

18 

2 

3 

I 

54 

05 

3 

4 

4 

5 

3 

13 

21 

57 

15 

16 

19 

4 

I 

32 

17 

Aug. 

5 

I 

20 

5 

2 

51 

18 

* 

21 

1 

52 

6 

I 

I 

32 

19 

2 

I 

26 

22 

2 

58 

7 

2 

I 

44 

3 

2 

30 

23 

3 

I 

10 

8 

3 

2 

13 

20 

4 

3 

37 

24 

4 

I 

39 

9 

4 

3 

41 

21 

5 

4 

53 

25 

5 

3 

08 

22 

6 

5 

I 

26 

23 

26 

I 

55 

10 

37 

24 

'7 

I 

28 

27 

2 

I 

02 

It 

I 

I 

. 8 

9 

2 

3 

32 

39 

28 

29 

3 

4 

I 

I 

15 

47 

12 

13 

2 

3 

1 

2 

50 

22 

25 

10 

4 

55 

30 

5 

3 

34 

14 

4 

4 

07 

26 

II 

5 

I 

30 

Oct. 

27 

28 

12 

I 

30 

I 

X 

59 

15 

42 

56 

29 

13 

2 

34 

2 

2 

I 

06 

16 

14 

3 

42 

3 

3 

I 

21 

17 

15 

4 

58 

4 

4 

I 

56 

18 

3 

31 

30 

16 

5 

I 

36 

5 

5 

4 

04 

19 

4 

4 

35 

3- 

Refraction 

Correction. 

Latitude  40°. 


h.  46" 
2 01 
2 40 
4 59 


1 50 

2 06 
2 49 
5 33 


1 54 

2 n 
2 59 
6 oi 


1 5S 

2 16 

3 04 
6 23 


2 00 

2,  19 

3 09 
6 38 


2 20 

I ” 

6 47 


I " 

6 49< 


2 00 

2 19 

3 09 

6 13 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS. 


TARLE  OF  LATITUDE  COEEFICIENTS. 


Latitude. 

Coefficient. 

Latitude. 

Coefficient. 

I.atitude. 

Coefficient. 

15“ 

•30 

30° 

•65 

45” 

1.20 

16 

•32 

31 

.68 

46 

1.24 

17 

•34 

32 

•71  1 

47 

1.29 

18 

•36 

33 

•75  1 

48 

r 33 

19 

.38 

34 

•78  1 

49 

1.38 

20 

.40 

35 

.82 

50 

1.42 

21 

.42 

36 

.85 

51 

1.47 

22 

.44 

37 

.89 

52 

1-53 

23 

.46 

38 

.92 

53 

1.58 

24 

.48 

39 

.96 

54 

1.64 

25 

•50 

40 

1 .00 

55 

1.70 

26 

•53 

41 

1.04 

56 

1.76 

27 

.56 

42 

1.08 

57 

1.82 

28 

•59 

43 

1 . 12 

58 

1.88 

29 

.62 

44 

1 . 16 

59 

1.94 

Note. — For  any  other  latitude  than  40°  the  refraction  corrections  given  in  the 
preceding  table  are  to  be  multiplied  by  the  coefficients  given  in  this  table  to  obtain 
the  true  refraction  corrections  for  that  latitude. 


latitude  corrections  we  find  that  the  refraction  corrections  will 
be  .94  of  those  given  in  the  table.  The  following  table  of 
declination  settings  may  now  be  made  out : 


Hour. 

Declination. 

Refr.  Cor. 

Setting. 

Hour. 

Declination. 

Refr.  Cor. 

Setting. 

7 

+ 6°  10'  54" 

+ 2'  00" 

+ 6°  12'  54" 

I 

+ 6°  16'  35" 

+ yi" 

4-  6°  17'  12" 

8 

6 II  51 

4-  I 10 

, 1 
0 13  01 

2 

6 17  31 

+ 41 

6 18  12 

9 

6 12  47 

+ 51 

6 13  38 

3 

6 18  28 

+ 51 

6 19  19 

10 

6 13  44 

+ 41 

6 14  25 

4 

6 19  25 

4-  i'  10" 

6 20  35 

II 

6 14  41 

+ 37 

6 15  18 

5 

6 20  22 

4-  2 00 

6 22  22 

From  March  20th  to  September  20th  the  declination  is 
positive,  while  from  September  20th  to  March  20th  it  is  nega- 
tive. From  December  20th  to  June  20th  the  hourly  correction 
is  positive,  while  from  June  20th  to  December  20th  it  is  nega- 
tive. The  refraction  correction  is  always  positive.  Particular 
attention  mu.st  be  given  to  all  these  signs  in  making  out  the 
table  of  declination  settings. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  49 


54.  Errors  in  Azimuth  due  to  Errors  in  the  Declina- 
tion and  Latitude  Angles. — The  spherical  triangle  involved 
in  an  observation  by  the  solar  compass  is  shown 
in  Fig.  II,  where  Pis  the  pole,  Z the  zenith,  and 
5 the  sun.  Then 

the  angle  at  P=t,  the  hour-angle  from  the 

meridian ; 

“ “ .2'=  the  azimuth  from  the  north 

point ; 

‘‘  S = q,  the  variable  or  parallactic 
angle. 

Also,  the  arc  PZ  =:  the  co-latitude  = 90°  — (p ; 

“ PS  — the  co-declination  = 90°  — 6 ; 

“ ZS  = the  co-altitude,  or  zenith  dis- 
tance = 90°  — h. 

Taking  the  parenthetical  notation  of  the  figure,  we  have, 
from  spherical  trigonometry, 

cos  (^)  = cos  [c]  cos  {U)  sin  {c)  sin  (^)  cos  (^). 

But  in  terms  of  d',  0,  and  A,  this  becomes 

sin  d = sin  0 sin  h — cos  0 cos  h cos  A.  . (i) 

In  a similar  manner,  from  two  other  fundamental  equations 
of  the  spherical  triangle,  we  may  write 

cos  d cos  t — cos  0 sin  h -f-  sin  0 cos  h cos  A ; (2) 

cos  d sin  / = cos  ^ sin  (3) 

If  we  differentiate  equation  (i)  with  reference  to  A and  d, 
4 


50 


SUR  VK  YING. 


and  then  with  reference  to  A and  0,  we  obtain,  after  some 
reductions  by  the  aid  of  equations  (2)  and  (3) 


and 


= 

dA^  — 


d d 

cos  0 sin  /’ 
dcf) 

cos  0 tan  { 


(4) 

(5) 


Now,  if  the  change  (or  error)  in  S and  0 be  taken  as  i minute 
of  arc,  or,  in  other  words,  if  the  settings  for  declination  or  lati- 
tude be  erroneous  by  that  amount,  either  from  errors  in  the 
instrumental  adjustments  or  othemise,  then  equations  (4)  and 
(5)  show  what  is  the  error  due  to  this  cause  in  the  azimuth,  or 
in  the  direction  of  the  meridian,  as  found.  In  the  following 
table,  values  of  dA^  and  dA^  are  given  for  various  values  of 
0 and  t (latitude  and  hour-angle).  In  this  table  no  attention 
is  paid  to  signs,  as  it  is  intended  mainly  to  show  the  size  of  the 
errors  to  which  the  work  is  liable  from  inaccurate  settings  for 
declination  and  latitude ; the  values  may,  however,  be  used  as 
corrections  to  the  observed  azimuths  from  such  inaccuracies  by 
observing  the  instructions  in  the  appended  note. 


* dAh  signifies  the  change  in  A due  to  a small  change,  d8,  in  d,  the  other 
functions  involved  in  equation  (i)  remaining  constant.  Similarly  for  dA^, 
when  <p  alone  changes.  The  derivation  of  equations  (4)  and  (5)  involves  a 
knowledge  of  the  infinitesimal  calculus. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  5 1 


TABLE  OF  ERRORS  IN  AZIMUTH  (BY  SOLAR  COMPASS)  FOR  i' 
ERROR  IN  DECLINATION  OR  LATITUDE. 


Houb. 

For  1'  Error  in  Declination. 

For  i'  Error  in  Latitude. 

Lat.  30° 

Lat.  40* 

Lat.  50® 

Lat,  30° 

Lat.  40® 

Lat.  50® 

11.30  A.M.  ) 

12.30  P.M,  )** 

8'.85 

lo'.o 

I2'.9 

8'.77 

9'.92 

ii'.8 

II  A.M.  j 
I P.M.  j 

f 

4.46 

5-05 

6 .01 

4.33 

4.87 

5 .80 

10  A.M.  i 
2 P.M.  ] 

1 

2 .31 

2 .61 

3-II 

2 .00 

2 .26 

2 .70 

9 A.M.  i 
3 P.M.  1 

i 

1 .63 

1.85 

2 .20 

I -15 

I .30 

1 .56 

8 A.M.  1 

4 P.M.  ) 

I -34 

I -51 

I .80 

0 .67 

0.75 

0 .90 

7 A.M.  i 
5 P.M.  1 

[ 

1 .20 

1 .35 

I .61 

0 .31 

0.35 

0.37 

6 A.M,  i 
6 P.M. 

i 

I 15 

1 .30 

I .56 

0.00 

0 .00 

0 .00 

Note, — Azimuths  observed  with  erroneous  declination  or  co-latitude  may  be 
corrected  by  this  table  by  observing  that  for  the  line  of  colli mation  set  too  high, 
the  azimuth  of  any  line  from  the  south  point  in  the  direction  S.W.N.E.  is 
found  too  small  in  the  forenoon  and  too  large  in  the  afternoon  by  the  tabular 
amounts  for  each  minute  of  error  in  the  altitude  of  the  solar  line  of  sight.  The 
reverse  is  true  for  this  line  set  too  low. 

Several  important  conclusions  may  be  drawn  from  this  table 
and  from  equations  (4)  and  (5). 

First — That  the  solar  compass  should  never  be  used  between 
II  A.M.  and  I P.M.,  and  preferably  not  between  10  A.M.  and  2 
P.M.,  if  the  best  results  are  desired. 

Second — That  at  6 o’clock  A.M.  and  P.M.,when  the  line  of  col- 
limation  lies  in  a plane  at  right  angles  to  the  plane  of  the  me- 
ridian, no  small  change  in  the  latitude-arc  will  affect  the  accu- 
racy of  the  result. 


52 


SURVEYING. 


Third — From  equation  (4)  and  (5),  we  see  that  both  errors 
have  the  same  sign.  Therefore,  if  the  declination-angle  be  er- 
roneously set  off,  and  the  latitude-angle  be  also  affected  by  mi 
equal  error  in  the  opposite  direction.,  then  the  two  resulting 
errors  in  azimuth  will  tend  to  compensate.  From  the  table  it 
may  be  seen  that  for  the  same  latitude  and  hour-angle  they 
would  nearly  balance  each  other  numerically.  If,  therefore, 
the  declination-angle  be  affected  by  an  error,  and  the  latitude 
of  the  place  then  found  by  a meridian  observation  zvith  the  com- 
pass, the  error  of  the  declination  zjuill  appear  in  the  resulting  lati- 
tude, zvith  the  opposite  sign.  In  this  way  any  constant  error 
in  the  declination-angle  may  be  nearly  eliminated. 

Fourth — The  best  times  of  day  for  using  the  solar  compass  are 
from  7 to  10  A.M.  and  from  2 to  5 P.M.  So  far  as  the  instru- 
mental errors  are  concerned,  the  greater  the  hour-angle  the 
better  the  observation ; but  when  the  sun  is  near  the  horizon, 
the  uncertainties  in  the  refraction  may  cause  unknown  errors 
of  considerable  size. 

Fifth — That  for  a given  error  in  the  setting  for  declination  or 
latitude  the  resulting  error  in  azimuth  will  have  opposite  signs 
in  forenoon  and  afternoon.  For,  in  equations  (4)  and  (5),  the 
hour-angle,  t,  has  different  signs  before  and  after  noon  ; and 
therefore  sin  t and  tan  t change  sign,  thus  changing  the  sign 
of  the  expression.  If,  therefore,  a lo-o’clock  azimuth  is  in 
error  5'  in  one  direction  from  erroneous  settings,  a 2-o’clock 
observation  with  the  same  instrument  should  give  an  azimuth 
5'  in  error  in  the  opposite  direction. 

55.  Solar  Attachments  are  appliances  fitted  upon  transit- 
instruments  for  the  purpose  of  finding  the  meridian,  the  same 
as  is  done  by  the  solar  compass.  The  principles  of  construc- 
tion and  use  are  the  same  as  those  of  the  solar  compass,  the 
application  of  these  principles  being  quite  various,  however, 
giving  rise  to  several  forms  of  attachments,  some  of  which  will 
be  discussed  in  connection  with  the  transit.  Their  adjust- 
ments and  limitations  are  nearly  the  same  as  those  here  given. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  $3 


EXERCISES  WITH  THE  SOLAR  COMPASS. 

56.  Determine  a true  meridian  line  by  an  observation  on  a circumpolar 
star  or  otherwise,  by  either  the  compass  or  transit.  Set  the  solar  compass  up 
on  one  point  of  this  line  with  a target  set  at  another  point  on  the  established 
meridian.  Having  carefully  adjusted  the  compass,  set  the  declination-arc  to  the 
right  angle  for  the  given  day  and  hour,  corrected  for  refraction,  and  make  a 
meridian  observation  on  the  sun  for  latitude.  If  the  true  latitude  of  the  place  is 
known,  the  difference  will  be  the  index  error  of  the  latitude-arc.  Leave  the  lati- 
tude-arc set  at  the  readings  obtained  by  the  meridian  observation  (whether  it  is 
the  true  latitude  or  not),  and  make  a series  of  determinations  of  the  meridian  by 
the  compass  at  various  times  of  day.  These  will  usually  be  in  error  from  the 
true  meridian  by  small  amounts.  Determine  the  size  of  these  errors  by  turn- 
ing upon  the  target  and  reading  the  horizontal  circle.  Record  these  errors, 
with  the  time  of  day  and  name  of  observer.  Each  student  should  make  a 
series  of  such  observations,  determining  for  himself  the  errors  to  which  the 
work  is  liable.  The  same  meridian  may  be  used  for  all,  after  it  has  been  prop- 
erly checked  by  duplicate  observations. 

57.  Set  the  latitude-  or  declination-angle  say  3'  from  its  true  value,  and 
observe  at  various  hours  of  the  day,  and  see  if  the  resulting  errors  in  azimuth 
are  about  three  times  those  given  in  the  table.  Note  that  these  resulting  errors 
are  in  opposite  directions  and  equal  in  amount  in  fore-  and  after-noon  observa- 
tions. 

58.  With  the  solar  compass  on  the  meridian  as  before,  select  a series  of 
points,  six  or  more,  which  are  fixed  and  plainly  visible  through  the  slits.  Find 
the  bearing  to  each  of  these  points  by  a separate  observation  on  the  sun  in 
each  case,  paying  no  attention  to  the  target  on  the  true  meridian.  Remove  the 
solar  compass  and  let  another  student,  ignorant  of  the  first  bearings,  set  the 
ordinary  needle  compass  over  the  same  point.  Bring  the  line  of  sight  upon 
the  target  and  make  the  needle  read  south  by  moving  the  vernier  on  the  decli- 
nation-arc. In  other  words,  set  off  the  declination  of  the  needle.  The  bear- 
ings given  by  the  needle-compass  should  now  agree  with  those  obtained  by  the 
solar  compass.  Read  upon  the  series  of  selected  points,  obtaining  the  bear- 
ings to  the  nearest  five  minutes.  Let  a third  student  take  a transit  (or  the  solar 
compass  would  do)  and  find  the  true  beacjngs  of  the  selected  points  with  refer- 
ence to  the  established  meridian.  Compare  results  and  so  obtain  some  data 
for  determining  the  relative  accuracy  of  the  solar  and  the  needle-compass. 
The  mean  of  two  azimuths  by  the  solar  compass  taken  on  the  same  line  at 
equal  intervals  from  noon  should  be  the  true  azimuth  of  the  line  if  the  instru- 
ment has  not  changed  its  adjustments  in  the  mean  time.  This  is  the  way  to 
find  the  true  azimuth  of  a line  by  the  solar  compass. 

59.  Run  a line  over  a series  of  points  (six  or  more)  in  the  forenoon  by  the 


54 


SURVEYING. 


solar  compass,  and  determine  the  bearings.  In  the  afternoon  run  it  back  again, 
using  the  bearings  obtained  in  the  forenoon,  set  other  stakes  where  the  points  are 
not  coincident  with  the  old  ones,  and  note  the  residual  discrepancy  at  the  close 
of  the  work.  Divide  this  by  twice  the  length  of  the  line,  and  this  is  the  error 
of  closure  due  to  erroneous  bearings.  The  chain  may  be  used  on  the  first  run- 
ning of  the  line,  but  on  the  retracing  the  stakes  may  be  set  opposite  the  first 
ones,  if  not  coincident.  The  object  is  to  determine  how  much  of  the  error  of 
closure  in  surveying  may  be  attributed  to  erroneous  bearings. 

Do  the  same  with  the  needle-compass  and  compare  results.  The  points 
need  not  be  in  line,  nor  need  they  enclose  an  area. 


ADJUSTMENT,  USE,  AND  CATE  OF  INSTRUMENTS.  55 


CHAPTER  III. 

INSTRUMENTS  FOR  DETERMINING  HORIZONTAL  LINES. 

PLUMB-LINE  AND  BUBBLE. 

6o.  The  Plumb-line  and  the  Bubble-tube  are  at  once  the 
most  simple,  universal,  and  essential  of  all  appliances  used  in 
surveying  and  astronomical  work.  Without  them  neither  the 
zenith  nor  the  horizon  could  be  effectually  determined,  and  the 
determination  of  altitudes  and  of  horizontal  lines  and  planes 
would  be  out  of  the  question.  Even  azimuths,  bearings,  and 
horizontal  angles  require  that  the  circle  by  which  they  are  ob- 
tained shall  be  brought  into  a horizontal  position. 

The  direction  of  the  plumb-line  is  by  definition  a vertical 
line,  pointing  to  the  zenith,  and  a plane  at  right  angles  to  this 
line  is  for  that  point  a horizontal  plane.  As  no  two  plumb- 
lines  can  be  parallel,  so  no  two  planes,  respectively  horizontal 
at  two  different  positions  on  the  earth’s  surface,  can  be  par- 
allel. 

Parallel  horizontal  planes  can  only  be  planes  at  different 
elevations,  all  horizontal  for  a single  position  on  the  earth’s 
surface. 

A level  surface  is  a surface  (not  a plane)  which  is  at  every 
point  perpendicular  to  a plumb-line  at  that  point.  If  the 
earth  were  covered  with  a fluid  in  a quiescent  state,  the  sur- 
face of  this  fluid  would  be  a level  surface.  This  surface  would 
not  be  a true  oblate  spheroid,  but  would  in  places  vary  several 
hundred  feet  from  such  a mean  spheroidal  surface.  This  is 
owing  to  the  fact  that  the  earth  is  not  a homogeneous  body. 


56 


SURVEYING. 


thus  causing  the  centre  of  mass  to  deviate  from  the  centre  of 
form.  Owing  also  to  much  irregularity  in  the  distribution  of 
the  mass,  with  respect  to  the  form  of  the  earth,  there  are  many 
irregular  deviations  of  the  plnmbdine'^  from  any  one  point.  A 
level  surface  follows  all  such  deviations. 

A bnbble-tube  is  a round  glass  tube  bent  or  ground  so  that 
its  inside  upper  surface  is  circular  on  a longitudinal  section. 
This  is  nearly  filled  with  ether,  the  remaining  space  being 
occupied  with  ether-vapor,  which  forms  the  bubble.  This 
tube  is  usually  graduated  to  assist  in  determining  the  exact 
position  of  the  bubble  in  the  tube.  If  the  tube  has  been 
ground  to  a perfect  circular  longitudinal  section,  then  a longi- 
tudinal line  tangent  to  this  inner  surface  at  the  centre  of  the  air- 
bubble  is  a level  line,  in  whatever  part  of  the  tube  the  bubble 
may  lie.  If  this  were  not  a level  line,  the  centre  of  gravity  of 
the  bubble  would  not  occupy  its  highest  possible  position  and 
would  move  until  it  did.  Since  the  position  of  the  centre  of 
a bubble  in  a tube  is  determined  by  reading  the  position  of  its 
ends  and  taking  the  mean,  it  is  necessary  that  the  arc  shall  be 
of  uniform  curvature — that  is,  circular. 

A line  tangent  to  the  inner  surface  of  the  bubble-tube  at 
its  centre,  as  defined  by  the  graduations  (or  another  line  parallel 
to  it)  is  called  the  axis  of  the  hibble.  When  the  bubble  is  in 
the  centre  of  the  tube,  therefore,  its  axis  is  horizontal. 

Proposition  /.  If  a bubble-tube  be  rigidly  attached  to  a 
frame,  and  if  this  frame  be  reversed  on  two  supports  lying  in 
the  vertical  plane  through  the  bubble-axis,  the  supporting 
points  are  level  when  the  bubble  occupies  the  same  portion  of 
the  tube  in  both  positions  of  frame,  whether  this  be  the  centre 


* In  the  northern  portion  of  the  United  States,  in  the  vicinity  of  the  Great 
Lakes,  deviations  of  the  plumb-line  (Clarke’s  Spheroid  being  used)  have  been 
found  as  great  as  lo  or  12  seconds  of  arc.  See  Primary  Triangulation  of  the 
U.  S.  Lake  Survey. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  $7 


or  not  ; providing,  of  course,  that  the  points  of  support  on  the 
frame  were  identical  in  the  two  cases. 

For,  the  tangent  horizontal  lines  being  identical  in  the  two 
positions  of  the  bubble,  the  vertical  distances  from  this  line  to 
the  points  of  support  must  be  equal,  otherwise  the  direct  and 
reversed  positions  would  not  give  identical  tangent  lines.  The 
points  of  support  are  therefore  in  a horizontal  line. 

Proposition  II.  If  a bubble-tube  be  revolved  about  an  axis 
in  such  a way  that  the  bubble  keeps  a constant  position  in  the 
tube,  the  axis  of  revolution  is  vertical. 

For,  since  the  bubble-tube  maintains  a constant  inclination 
to  the  horizon  (this  inclination  being  zero  when  the  bubble  is 
in  the  centre),  the  plane  of  motion  can  have  no  vertical  com- 
ponent, and,  therefore,  the  axis  of  revolution  must  be  vertical. 

Cor.  I.  Similarly  we  may  say  that  if  a bubble-tube  be  re- 
volved i8o°  about  an  axis,  and  if  the  bubble  have  the  same 
reading  in  the  two  positions,  then  the  plane  of  revolution  has 
no  vertical  component  in  the  direction  of  the  bubble-axis,  and 
therefore  the  axis  of  revolution  lies  in  a vertical  plane  at  right 
angles  to  the  bubble-axis.  If  the  same  test  be  made  for  any 
other  two  horizontal  positions  i8o°  apart  (preferably  90°  from 
first  position)  and  the  bubble  have  the  same  reading  in  the 
two  cases,  then  the  axis  of  revolution  lies  in  a vertical  plane  at 
right  angles  to  these  new  positions  of  bubble-axis,  and  there- 
fore it  lies  at  the  intersection  of  these  two  vertical  planes,  or  it 
is  vertical.  If  two  bubble-tubes  (not  parallel  to  each  other 
and  preferably  at  right  angles)  be  rigidly  attached  to  a frame 
that  revolves  about  an  axis,  and  if  each  bubble  has  the  same 
reading  in  two  positions  of  frame  180°  apart,  the  axis  of  revo- 
lution is  vertical,  even  though  the  two  bubbles  do  not  read 
alike  nor  either  is  at  the  middle  of  its  tube. 

Cor.  2.  In  all  cases  where  a bubble-tube  has  been  shifted 
180°  in  the  same  supports,  or  axis,  the  angular  difference 
between  the  two  positions  of  the  bubble  is  twice  the  angular 


58 


SURVEYING. 


deviation  of  the  supports  from  a horizontal,  or  of  the  axis  from 
a vertical. 

61.  The  Accurate  Measurement  of  Small  Vertical  An- 
gles is  accomplished  by  means  of  the  bubble  with  greater  read- 
iness and  precision  than  by  any  other  device  known.  For  this 
purpose  the  bubble  should  be  ground  accurately  to  the  arc  of 
a circle  with  a long  radius,  and  uniformly  graduated,  d'hen  a 
given  bubble-movement  in  any  part  of  the  tube  corresponds  to 
a known  angular  change,  when  the  angular  value  of  a move- 
ment of  one  division  in  the  graduated  scale  has  been  deter- 
mined. These  graduations  are  usually  made  on  the  top  of  the 
glass  tube.  To  measure  a small  angle  by  means  of  the  bubble, 
read  the  two  ends  of  the  bubble  to  divisions  and  tenths,  and 
take  the  one  half-difference  of  end  readings.*  Shift  the  bubble 
a given  amount  and  read  both  ends  again,  taking  one  half  the 
difference.  The  difference  of  these  two  results  in  divisions  of 
the  scale,  multiplied  by  the  angular  value  of  one  division  on 
the  scale,  is  the  vertical  angle  through  which  the  tube  was 
shifted. 

62.  The  Angular  Value  of  One  Division  of  the  Bubble 

may  be  found  in  various  ways. 

{a)  By  a telescopic  line  of  sight.  Attach  the  bubble-tube  rig- 
idly to  a mounted  telescope,  putting  the  bubble-axis  in  the  plane 
of  the  telescope.  Measure  off  a convenient  base-line  on  level 
ground  of  from  200  to  500  feet.  Set  the  telescope  at  one  end 
of  this  base,  and  hold  a rod  vertically  at  the  other.  Bring  the 
bubble  near  one  end  of  its  tube  by  moving  the  telescope  verti- 
cally, and  read  the  two  ends.  Read  the  height  of  the  cross- 
wires on  the  rod.  Bring  the  bubble  near  the  other  end  of  tube 
and  read  both  the  bubble  and  rod.  Repeat  many  times.  Re- 
duce the  work  by  taking  the  half-difference  of  the  two  end 

* Rubbles  are  read  from  the  middle  outwards  towards  the  ends.  Then  the 
half-difference  of  end  readings  is  the  distance  of  the  centre  of  the  bubble  from 
the  centre  of  the  scale. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  59 


readings  in  each  case,  thus  giving  the  distance  of  the  centre  of 
the  bubble  from  the  centre  of  tube  for  each  position.  Take 
the  mean  of  these  results  for  each  set  of  end  readings  sepa- 
rately. If  these  mean  results  were  for  opposite  ends  of  the 
tube,  add  them  together  and  this  gives  the  average  movement 
of  bubble.  Similarly  take  the  mean  of  the  upper  readings  and 
the  mean  of  the  lower  readings  on  the  rod,  and  take  the  differ- 
ence, and  this  is  the  average  movement  of  the  line  of  sight. 
Calling  the  bubble-movement  in  divisions  of  scale  D,  the  move- 
ment on  the  rod,  in  feet,  and  the  length  of  the  base,  in  feet, 
B.,  we  would  have,  in  seconds  of  arc, 

angular  value  of  i div.  of  bubble  = • — 77« 

^ BD  sm 

{b)  By  a large  vertical  circle.  Mount  the  bubble  rigidly 
upon  the  circle,  having  its  axis  parallel  to  the  plane  of  the 
circle.  Move  the  bubble  from  end  to  end  of  tube,  as  before, 
reading  the  corresponding  angular  changes  directly  upon  the 
circle.  Divide  the  mean  angular  movement  by  the  mean 
movement  of  bubble. 

This  requires  a large  circle  with  micrometer  attachments, 
such  as  is  used  on  astronomical  instruments. 

(c)  By  a level  trier.  This  consists  of  a beam  hinged  at  one 
end  and  moved  vertically  by  means  of  a micrometer  screw  at 
the  other.  The  bubble-tube  is  placed  upon  the  beam,  and  the 
bubble  moved  back  and  forth  by  means  of  the  screw,  each 
revolution  of  which  gives  a known  angular  movement  to  the 
beam. 

63.  General  Considerations. — A bubble  is  sensitive  direct- 
ly as  the  length  of  the  radius  of  curvature,  or  indirectly  as  its 
rate  of  curvature.  It  is  also  sensitive  in  proportion  to  its 
length,  a long  bubble*  settling  much  more  quickly  and  ac- 


This  refers  to  the  length  of  the  air-bubble  itself,  and  not  to  the  glass  tube. 


6o 


SURVEYING. 


curately  than  a short  one.  Some  bubble-tubes  have  a cham- 
ber at  one  end  connected  with  the  main  space  by  a small  hole 
through  the  bottom  of  the  dividing  partition.  This  enables 
the  length  of  the  bubble  to  be  under  control.  As  ether  ex- 
pands and  contracts  very  largely  with  temperature,  the  bubble 
is  apt  to  be  too  long  in  winter  and  too  short  in  summer  if  the 
chamber  is  not  used.  The  bubble-tube  should  not  be  rigidly 
confined  by  metallic  fastenings  about  its  centre,  if  the  value  of 
one  division  is  significant,  as  the  changes  of  temperature  will 
change  its  curvature.  Bubble-tubes,  or  level-vials  as  they  are 
often  called,  may  be  sealed  by  glass  stoppers  set  in  a glue 
made  by  dissolving  isinglass  in  hot  water,  and  covering  with 
gold-beater’s  skin  set  with  the  same  glue,  the  whole  varnished 
over  when  dry. 


THE  engineer’s  LEVEL. 

64.  The  Engineer’s  Level  consists  of  a telescopic  horizon- 
tal line  of  sight  joined  to  a spiiit-level,  the  whole  properly 
supported  and  revolving  on  a vertical  axis.  Such  an  instru- 
ment is  shown  in  Fig.  12.  The  vertical  parts  of  the  frame 
which  support  the  telescope  are  called  wyes,  and  the  cylindri- 
cal bearings  on  the  telescope-tube  are  called  the  pivot-rings. 
The  telescope  can  be  lifted  out  of  the  wyes  by  loosening  the 
clips  over  the  rings,  these  being  held  by  the  small  pins 
attached  to  strings  and  shown  in  the  cut.  A clamp  and 
tangent-screw  are  connected  with  the  axis  for  holding  it  to  a 
given  pointing  or  for  moving  it  horizontally  while  clamped. 
The  attached  bubble  enables  the  line  of  sight  in  the  telescope 
to  be  brought  into  a horizontal  position. 

The  construction  of  the  instrument  is  best  shown  by  the 
sectional  view  given  in  Fig.  13. 

The  objective  is  a compound  lens,  the  two  parts  having 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  6l 


different  refractive  powers  in  order  that  the  image  may  be 
flat.  A simple  lens  gives  a spherical  image.  The  image  is 


formed  at  the  plane  of  the  cross-wires,  which  are  attached  to  the 
reticule  held  in  place  by  the  capstan-screws  shown  in  the  cut.  The 


62 


SUR  VE  YING. 


line  of  sight,  or  line  of  collimation,  is  the  line  joining  the  two 
corresponding  points  in  object  and  image  with  which  the 


ADJUSTMENT,  USE,  AND  CADE  OF  INSTRUMENTS.  63 


intersection  of  the  cross-wires  coincides.'^  Evidently  this 
line  of  sight  may  lie  anywhere  in  the  field  of  view  within  the 
limits  of  movement  of  the  reticule.  The  eye-piece  serves  only 
to  magnify  the  image,  and  sometimes  to  invert  it,  as  is  the 
case  in  the  sectional  view  of  Fig.  13.  The  image  itself  is 
always  inverted  ; and  if  this  be  examined  by  an  eye-piece  of 
two  lenses,  which  simply  magnifies  but  does  not  invert,  the 
object  is  seen  in  an  inverted  position.  If  four  lenses  are  used 
in  the  eye-piece,  it  re-inverts  the  image  so  that  the  object  is 
seen  erect.  This  results  in  a loss  of  light  and  of  distinctness. 

ADJUSTMENTS  OF  THE  LEVEL. 

65.  To  make  the  Line  of  Sight  parallel  to  the  Axis  of 
the  Bubble. 

First,  or  Indirect,  Method. — This  method  rests  on  the 
proposition  that  if  two  lines  are  parallel  to  a third  line,  they  are 
parallel  to  each  other.  This  method  is  indirect,  but  the 
manipulations  are  readily  performed.  It  is  the  usual  method,, 
and  is  frequently  given  as  two  separate  adjustments. 

First,  bring  the  line  of  sight  to  coincide  with  the  centres  of 
the  pivot-rings  by  revolving  the  telescope,  bottom  side  up,,  in 
the  wyes,  and  adjusting  the  reticule  until  the  intersectio^n  of 
the  wires  remains  on  a fixed  point  of  the  image. f If  the 

* More  correctly,  it  is  the  line  joining  the  inner  pHncipal  point  of  the  objec- 
tive with  that  point  of  the  image  covered  by  the  intersection  of  the  cross-wires. 
See  Fig,  61,  and  note  to  same. 

f The  optical  axis  of  a lens  is  the  line  joining  the  centres  of  the  Irwe  spherical 
surfaces  bounding  it.  If  this  axis  is  not  coincident  with  the  axis  of  the  tele- 
scope, or  rings,  owing  to  an  erroneous  adjustment  of  the  objective  slide  by  the 
screws  near  the  centre  of  the  telescope  tube,  Fig.  13,  or  the  improper  setting  of 
the  lens  in  its  case,  then  the  image  will  be  shifted  laterally  a small  amount  equal 
to  the  lateral  deviation  of  the  two  ‘ ‘ principal  points”  of  the  lens  from  each  other. 
In  this  case  the  image  itself  will  appear  to  rotate  as  the  telescope  is  revolved. 
If  now  the  centre  of  the  cross-wires  be  moved  so  as  to  remain  on  a fixed  portion 
of  the  image,  it  no  longer  occupies  the  axis  of  the  telescope,  but  the  line  of  sight 
is  now  parallel  to  this  axis,  so  that  this  adjustment  still  accomplishes  all.  that  is 


64 


SURVEYING. 


instrument  gives  an  erect  view  of  the  object,  there  is  one 
inversion  between  the  wires  and  the  eye,  and  therefore  tlie 
reticule  must  be  moved  in  the  direction  of  and  one  half  the 
amount  of  its  apparent  displacement.  If  the  view  is  inverted, 
there  is  no  inversion  between  wires  and  eye,  and  therefore  its 
apparent  is  its  true  displacement. 

Second,  make  the  axis  of  the  bubble-tube  parallel  to  the 
bottoms  of  the  rings  by  reversing  the  telescope  end  for  end 
in  the  wyes  and  adjusting  the  bubble  until  it  remains  in  the 
centre  of  the  tube  for  the  two  positions.  The  telescope 
should  be  removed  and  replaced  with  great  care  so  as  not  to 
disturb  the  relative  elevation  of  the  wyes  by  any  jar  or  shock. 
The  axis,  of  course,  should  be  clamped  to  prevent  any  hori- 
zontal motion  in  making  either  part  of  this  adjustment. 

This  method  is  based  on  an  assumption  which  may  or  may 
not  be  true.  It  is  that  the  pivot-rings  are  of  the  same  size, 
and  therefore  the  lines  joining  their  centres  and  bottoms  are 
parallel. 

To  find  the  relative  size  of  the  pivot-rings,  use  a striding- 
level  resting  on  the  two  pivot-rings  and  read  in  reversed  posi- 
tions. Then  change  the  rings  in  their  supports  and  read  the 
level  again  in  reversed  positions.  To  reduce  the  notes,  the 
value  of  one  division  of  the  striding-level  must  be  known.* * 

The  objective  is  always  properly  centred  and  adjusted  when 
the  instrument  leaves  the  maker’s  hands;  but  it  is  apt  to 
become  loose  in  its  frame,  and  this  frame  also  loosens  in  the 
telescope-tube.  If  the  glass  is  loose  in  its  frame,  unscrew  it 
from  the  telescope-tube  and  screw  up  the  tightening  band 


desired.  Or,  the  objective  may  have  its  optical  axis  coincident  with  that  of  the 
telescope  and  the  optical  axis  of  the  eye-piece  not  parallel  to  that  of  the  objec- 
tive, and  this  will  cause  the  image  and  wires  to  appear  to  rotate  together — when 
the  telescope  is  revolved.  This  need  cause  no  error  in  the  work,  but  should  be 
adjusted  by  the  screws  shown  just  back  of  the  capstan  screws.  Fig.  13. 

* See  adjustments  in  Precise  Levelling,  Chap.  XIV. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  65 
I — 

from  the  rear  side.  Do  not  take  the  glasses  apart  tinder  any 
circumstances,  for  they  are  ground  for  a given  relative  position 
and  would  not  be  true  for  any  other.  A loose  objective  is  a 
fatal  defect  in  a levelling-instrument  and  must  be  constantly 
guarded  against. 

Second,  or  Direct,  Method. — This  consists  in  adjusting  the 
bubble  directly  to  the  line  of  sight,  whether  this  be  in  the  cen- 
tre of  the  pivot-rings  or  not.  It  is  sometimes  called  the 
peg  adjustment.’'  Drive  two  pegs  on  nearly  level  ground 
about  200  feet  apart.  Set  the  level  about  eight  or  ten  inches 
from  one  of  them,  or  so  that  when  the  rod  is  held  upon  it  in  a 
vertical  position  the  eye  end  of  the  telescope  will  swing  about  a 
half  inch  from  its  face.  Turn  the  eye  end  of  the  telescope  upon 
the  graduated  face  of  the  rod,  the  bubble  being  in  the  middle  of 
its  tube;  look  through  the  object  end  and  set  a pencil-point  on 
the  rod  at  the  centre  of  the  small  field  of  view,  which  should 
be  from  i to  J inch  in  diameter.  Read  the  elevation  of  this 
point,  which  we  will  call  a.  Hold  the  same  rod  on  the  distant 
peg  and,  with  the  bubble  in  the  middle,  set  the  target  on  the 
line  of  sight,  and  call  this  reading  b.  Now  carry  the  instru- 
ment to  the  distant  peg,  set  it  near  it,  read  the  elevation  of  the 
instrument  as  before,  which  reading  we  will  call  a'  ; carry  the 
rod  to  the  first  peg  and  set  the  target  on  the  line  of  sight,  giv- 
ing the  reading  b' . If  the  line  of  sight  had  been  parallel  to  the 
axis  of  the  bubble  in  each  case,  it  would  have  been  horizontal 
when  the  bubble  was  in  the  middle  of  the  tube,  and  hence  the 
difference  between  the  a and  b readings  in  each  case  would 
have  been  the  difference  of  elevation  of  the  pegs.*  We 
should  therefore  have  had 

a-b  = d^a^ (i) 


*This  assumption  neglects  the  effect  of  the  earth’s  curvature.  This  is 
eight  inches  to  one  mile,  and  is  proportional  to  the  square  of  the  distance.  For 
200  feet  it  would  be  about  0.001  of  a foot,  and  twice  this,  or  0.002  of  a foot,  is 
the  error  made  in  the  above  assumption. 

5 


66 


SUR  VE  YING. 


If  the  line  of  sight  was  not  parallel  to  the  axis  of  the  bub- 
ble, however,  then  the  differences  of  elevation  of  the  two  pegs, 
as  obtained  by  the  two  sets  of  observations,  are  not  equal,  and 
we  should  have 

{a-D-ib'  -a')  = d (2) 

Now  d is  twice  the  deviation  of  the  line  of  sight  from  the 
bubble-axis  for  the  given  distance.  (Let  the  student  construct 
a figure  and  show  this.)  If,  therefore,  the  target  be  moved 
up  or  down  as  the  case  may  be,  a distance  equal  to  \d,  then 
the  line  of  sight  may  be  brought  to  this  position  by  the 
levelling-screws,  and  the  bubble  adjusted  to  bring  it  to  the 
middle,  or  else  the  instrument  may  be  left  undisturbed  with 
the  bubble  in  the  middle,  and  the  line  of  sight  adjusted  to 
read  upon  the  target  by  moving  the  reticule.  The  significant 
fact  is  that  by  moving  the  target  \d  from  its  last  position  a 
true  horizontal  line  is  established,  and  either  the  bubble  or  the 
line  of  sight  can  be  adjusted  to  it  after  the  other  has  been 
brought  into  a horizontal  position  by  means  of  the  levelling- 
screws.  Equation  (2)  may  be  written 

{a  — {b  b')  — d\ (3) 

from  which  it  may  be  seen  at  once  that  the  line  of  sight 
inclines  down  when  d is  positive,  and  up  when  d is  negative. 
We  may  therefore  have  for  setting  the  target  the  following 

Rule:  Add  together  the  two  heights  of  instrument  and  the 
two  rod  readings,  subtract  the  latter  from  the  former,  and  take 
one  half  the  remainder.  Move  the  target  by  this  amount  from 
the  b'  reading,  up  when  positive  and  down  when  negative.  It  is 
then  in  a horizontal  line  with  the  cross-wires  of  the  instrument. 

It  will  be  noted  that  no  distances  are  measured  in  the  above 
method  as  is  usually  prescribed  in  peg-adjustments.  After 
adjusting  either  the  line  of  sight  or  the  bubble  at  the  second 
peg,  return  to  the  first  peg,  read  height  of  instrument  again, 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  67 


and  then  read  the  rod  on  the  second  peg  for  a check.  See  if 
this  new  value  of  {a  — b)  agrees  with  the  adjusted  value  of 
{b'  — a').  If  not,  adjust  again. 

This  method  is  independent  of  the  relative  size  of  the  pivot- 
rings  and  of  the  condition  of  the  objective.  (The  objective 
must  have  a fixed  condition  or  no  adjustment  is  worth  any- 
thing.) Although  the  essential  relation  of  parallelism  is  ob- 
tained between  the* line  of  sight  and  the  bubble,  it  must  not 
be  expected  that  the  telescope  can  be  reversed  in  the  wyes  or 
revolved  180°  about  its  axis  without  both  these  auxiliary 
adjustments  appearing  to  be  in  error.  For  inasmuch  as  these 
two  lines  have  been  made  parallel  without  reference  to  the 
axis  of  the  telescope  or  to  the  bottoms  of  the  rings,  they 
probably  are  not  parallel  to  either  of  these.  If  the  first  meth- 
od is  used  and  the  adjustment  made,  it  should  stand  the  test 
of  the  second  (the  necessary  assumptions  being  true),  but  if 
adjusted  by  the  second  method  it  should  not  be  expected  to 
stand  the  test  of  the  first  method.  At  the  same  time  the 
second  method  is  absolute,  while  the  first  is  based  on  assump- 
tions that  are  often  untrue.  This  adjustment  should  be  exam- 
ined every  day  in  actual  practice. 

66.  To  bring  the  Bubble-axis  into  the  Vertical  Plane 
through  the  Axis  of  the  Telescope. — Turn  the  telescope 
slightly  back  and  forth  in  the  wyes,  and  note  the  action  of  the 
bubble.  If  it  remains  in  the  centre  the  adjustment  is  correct. 
If  not,  move  one  end  of  the  bubble  by  means  of  the  lateral 
adjusting-screws.  If  this  adjustment  is  very  much  in  error  it 
should  be  made  approximately  right  before  going  on  with  the 
preceding  adjustment. 

67.  To  make  the  Axis  of  the  Wyes  perpendicular  to 
the  Vertical  Axis  of  the  Instrument. — This  is  to  enable  the 
telescope  to  be  revolved  horizontally  without  re-levelling. 
Level  the  instrument  in  one  position.  Revolve  180°  horizon- 
tally, and  correct  one  half  the  movement  of  the  bubble  by  the 


68 


SUK  VE  YING. 


wye-adjustment  and  the  other  half  by  the  levelling-screws. 
Repeat  for  a check. 

68.  Relative  Importance  of  Adjustments. — The  first 
adjustment  is  by  far  the  most  important.  The  second  can 
only  enter  in  the  work  when  the  telescope  is  revolved  slightly 
from  its  true  position  in  the  wyes.  Most  modern  levels  have 
some  device  for  holding  the  telescope  in  its  proper  position 
when  in  use.  This  position  is  such  as  brings  the  horizontal 
wire  truly  horizontal.  The  last  adjustment  given  is  only  a 
matter  of  convenience.  It  saves  stopping  to  relevel  after  re- 
volving the  telescope.  It  docs  not  affect  the  accuracy  of  the 
work  appreciably.  It  is  absolutely  essential,  however,  that  the 
line  of  sight  should  be  truly  horizontal  when  the  bubble  is  in 
the  middle  of  the  tube,  or  reads  zero,  and  this  makes  the  first 
adjustment  here  given  of  such  vital  consequence. 

69.  Focussing  and  Parallax. — The  eye-piece  serves  to 
give  a distinct  and  magnified  view  of  the  image.  It  also  inverts 
the  image  in  all  instruments  where  the  object  is  seen  in  an 
erect  position.  Since  the  magnifying  power  of  the  eye-piece 
is  large,  its  focal  range  of  distinct  vision  is  very  small,  depend- 
ing on  its  magnifying  power.  With  the  ordinary  field-instru- 
ments it  is  about  one  sixteenth  of  an  inch.  Both  the  virtual 
image,  as  formed  by  the  objective,  and  the  cross-wires,  should 
lie  in  the  focus  of  the  eye-piece.  They  should  therefore  lie  in 
the  same  plane.  Now  the  virtual  image  may  be  moved  back 
and  forth  by  moving  the  objective  in  or  out,  but  the  plane  of 
the  cross-wires  is  fixed.  If  the  two  are  brought  into  the  same 
plane,  therefore,  the  image  must  be  brought  upon  the  wires. 
To  accomplish  this,  first  focus  the  eye-piece  on  the  wires  so  that 
they  appear  most  distinct.  In  doing  this  there  should  be  no 
image  visible,  so  that  either  the  objective  is  thrown  out  of 
focus  or  the  telescope  is  turned  to  the  sky.  The  eye-piece  is 
most  accurately  focussed  by  finding  its  inner  and  outer  limits 
for  distinct  vision  of  the  wires,  and  then  setting  it  at  the  mean 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  69 


position.  The  objective  may  now  be  moved  until  the  image 
also  comes  into  focus.  This  will  have  to  be  done  for  each 
pointing  if  the  distances  are  different.  If  the  image  is  not 
brought  into  exact  coincidence  with  the  cross-hairs,  these  will 
seem  to  move  slightly  on  the  image  as  the  eye  is  moved  behind 
the  eye-piece.  This  angular  displacement  of  the  wires  on  the 
image  is  called  parallax^  and  can  only  occur  when  they  are  not 
in  the  same  plane.  It  is  removed  by  refocussing  the  object- 
ive, thus  moving  the  image,  until  there  is  no  perceptible  rela- 
tive movement  of  wires  and  image  as  the  eye  is  shifted,  when 
they  are  practically  in  coincidence.  If  there  is  parallax,  the 
reading  may  be  in  error  by  its  maximum  angular  amount.  If 
the  eye  were  always  held  at  the  centre  of  the  eye-piece  there 
would  be  no  parallax,  and  it  is  to  accomplish  this  that  the  eye- 
piece is  covered  by  a shield  with  a small  hole  in  its  centre. 
Still,  the  slight  movement  of  the  eye  thus  allowed  is  sufficient 
to  cause  some  parallactic  error  if  the  wires  and  image  are  not 
practically  coincident.  When  the  eye-piece  is  once  adjusted  to 
distinct  vision  on  the  cross-wires  it  requires  no  further  atten- 
tion so  long  as  the  instrument  is  used  by  the  same  person. 
Another  person,  having  eyes  of  a different  focal  range,  would 
have  to  readjust  the  eye-piece.  The  eye-piece  adjustment, 
therefore,  is  personal,  and  is  made  once  for  all  for  a given  indi- 
vidual; while  the  objective  adjustment  depends  on  the  dis- 
tance of  the  object  from  the  instrument,  is  made  for  each 
pointing,  and  is  considered  perfect  when  the  parallax  is  re- 
moved.* 


* This  discussion  is  worded  for  an  erecting  telescope,  where  the  objective 
moves.  In  an  inverting  instrument  the  eye-piece  and  reticule  may  move  together 
in  the  telescope  while  the  objective  remains  fixed.  Here  the  image  takes  differ 
ent  positions  in  the  telescope-tube,  as  the  distances  vary,  and  the  cross-wires 
are  moved  to  suit.  There  is  also  a motion  of  the  eye-piece  with  reference  to 
the  wires,  and  this  is  the  eye-piece  adjustment  ; while  the  movement  of  both 
together  is  what  is  called  the  objective  adjustment  in  the  above  discussion. 


70 


SUR  VE  YING. 


Fig.  14. 


Fig.  15. 


THE  LEVELLING-ROD. 

70.  The  Levelling-rod  is  used  to 
measure  tlic  vertical  distance  from  the 
line  of  siglit  down  to  the  turning-point 
or  bench-mark.  There  are  two  general 
classes,  Self-reading,  or  Speaking,  and 
Target  Rods. 

A Self-reading,  or  Speaking,  Rodis 
one  so  graduated  as  to  enable  the  ob- 
server to  note  at  once  the  reading  of  the 
point  which  lies  in  the  line  of  sight,  this 
reading  being  in  all  cases  the  distance 
to  the  bottom  of  the  rod.  The  rod- 
man  here  has  nothing  to  do  but  to  hold 
the  rod  vertical.  The  observer  notes 
and  records  the  reading. 

A Target-rod  is  furnished  with  a 
sliding  target  moved  by  the  rodman  in 
response  to  signals  from  the  observer 
until  it  accurately  coincides  with  the 
line  of  sight.  Its  position  is  then  read 
with  great  accuracy  by  means  of  a ver- 
nier scale. 

Fig.  14  is  one  form  of  self-reading 
rod  which  is  also  fitted  with  a target. 
This  is  called  the  Philadelphia  rod. 
Fig.  15  is  the  New  York  rod  and  is  not 
self-reading.  It  is  the  standard  target- 
rod  used  in  this  country.  The  one 
here  shown  is  in  three  sections,  whereas 
those  in  common  use  are  in  two  parts 
only. 

Various  other  patterns  of  self-read- 
ing rods  are  used.  For  rough  work  a 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  7 1 


twelve-  or  fourteen-foot  rod,  2 inches  wide  and  inches  thick, 
painted  and  fitted  with  an  iron  or  brass  shoe  at  bottom,  gradu- 
ated to  hundredths  of  a foot,  will  be  found  very  efficient.  The 
graduations  should  be  so  distinct  that  they  can  be  read  through 
the  telescope  at  a distance  of  five  or  six  hundred  feet. 

THE  USE  OF  THE  LEVEL. 

71.  The  Level  is  used — 

{a)  To  find  the  relative  elevation  of  points  a considerable 
distance  apart. 

(^)  To  obtain  the  profile  of  a line. 

(c)  To  establish  a grade. 

These  objects  may  be  more  or  less  intermingled  in  any 
given  piece  of  work.  Whatever  may  be  the  ultimate  object  of 
the  work,  however,  the  immediate  object  for  any  given  setting 
of  the  instrument  is  to  find  how  much  higher  or  lower  a certain 
forward,  or  unknown,  point  is  than  a certain  other  back,  or 
known,  point.  Thus,  the  rod  being  held  on  the  known  point, 
the  line  of  sight  is  turned  upon  it  and  the  rod-reading  gives  at 
once  the  height  of  instrument  above  that  point.  If  the  rod  be 
now  held  on  the  forward,  or  unknown,  point,  and  the  line  of 
sight  turned  upon  it,  this  rod-reading  gives  the  distance  of  that 
point  below  the  line  of  sight.  The  reading  on  the  known 
point  is  called  the  back-sight,  and  that  on  the  unknown  point 
is  called  the  foresight.  If  the  elevation  of  the  known  point 
be  given,  we  find  the  elevation  of  the  line  of  sight  by  adding 
the  rod-reading  at  that  point.  By  subtracting  from  this  eleva- 
tion the  reading  on  the  unknown  point,  the  elevation  of  that 
point  is  obtained.  Thus  we  have  found  the  relative  elevations 
of  the  two  points  by  referring  them  both  to  the  horizontal 
plane  through  the  instrument.  Since  the  back-sight  reading 
gives  the  elevation  of  the  instrument,  and  since  this  is  always 
greater  than  the  elevation  of  that  point,  it  follows  that  the 
back-sight  reading  is  essentially  positive.  For  a similar  reason 


72 


S UR  VE  Y I NG. 


the  fore-sight  reading  is  essentially  negative,  since  any  point 
on  which  the  rod  is  held  is  lower  than  the  line  of  sicrht. 

o 

It  will  also  be  seen  that  there  can  be  but  one  back-sight  (un- 
less the  height  of  the  instrument  is  to  be  found  from  readings 
on  several  known  points,  and  the  mean  taken),  while  there  can 
be  any  number  of  fore-sights  from  one  instrument  position. 
Thus,  the  height  of  the  instrument  having  been  determined, 
the  elevations  of  any  number  of  points,  in  any  direction,  may 
be  determined  by  referring  them  all  to  the  horizontal  plane 
through  the  instrument,  whose  elevation  has  been  obtained  by 
the  single  back-sight  reading.  It  is  also  important  to  remem- 
ber that  the  terms  “ back-sight”  and  “ fore-sight”  have  no 
reference  to  directions  or  points  of  the  compass,  but  they  do 
have  a rational  significance  when  we  think  of  the  work  pro- 
ceeding from  the  known  point  to  the  unknown  point  or  points. 
Thus,  w'e  refer  back  to  the  known  point  for  height  of  instru- 
ment, and  then  transfer  this  knowledge  forward  to  the  points 
whose  elevations  we  wish  to  find. 

DIFFERENTIAL  LEVELLING. 

72.  Differential  Levelling  consists  in  finding  the  differ- 
ence of  elevation  of  points  a considerable  distance  apart.  The 
elevation  of  the  first  point  being  known  or  assumed,  the  differ- 
ence of  elevation  between  this  and  any  other  point  is  found 
and  added  algebraically,  thus  giving  the  elevation  of  the  second 
point.  The  “plane  of  reference”  is  the  surface  of  zero-eleva- 
tion and  is  generally  called  the  “datum  plane.”  This  is  not 
really  a plane  but  a level  surface,  according  to  the  definition 
given  in  art.  60.  It  is,  however,  universally  denominated  the 
“plane  of  reference,”  “datum  plane,”  or  simply  “datum.” 
The  problem,  then,  is  to  find  the  difference  of  elevation  between 
two  distant  points.  If  the  points  were  near  together  and  had 
not  too  great  a difference  of  elevation,  a single  setting  of  the 
instrument  would  be  sufficient.  If  they  are  too  far  apart  for 


ADJUSTMENT,  USE,  AND  CATE  OF  INSTRUMENTS.  73 


this,  either  in  distance  or  in  elevation,  then  more  than  one 
setting  of  the  instrument  must  be  made.  In  this  case  the 
intervening  points  occupied  by  the  rod  are  called  turyiing-points, 
the  terminal  points  being  called  bench-marks.  The  successive 
differences  of  elevation  of  these  turning-points  is  determined 
by  setting  the  level  equally  distant  from  them,  and  so  they 
serve  to  divide  up  the  total  distance  between  terminal  points 
into  a series  of  short  spaces,  each  of  which  can  be  covered  by 
a single  setting  of  the  instrument.  The  successive  differences 
of  elevation  of  turning-points  being  found,  their  algebraic  sum 
would  be  the  difference  of  elevation  of  the  terminal  points,  or 
bench-marks.  But  since  all  the  back-sights  are  essentially 
positive  and  all  the  fore-sights  are  essentially  negative,  we  may 
at  once  add  all  the  back-sights  together  and  all  the  fore-sights 
together,  and  take  the  difference  of  the  sums.  This  is  the 
difference  of  elevation  between  terminal  points,  and  has  the 
sign  of  the  larger  sum,  the  back-sights  being  positive  and  the 
fore-sights  negative.  This  difference  of  elevation  added  alge- 
braically to  the  elevation  of  the  initial  point  gives  the  elevation 
of  the  final  point.  Evidently  the  route  travelled  in  passing 
from  one  bench-mark  to  another  is  of  no  consequence  so  long 
as  the  true  difference  of  elevation  is  obtained. 

73.  Length  of  Sights. — Where  the  ground  is  nearly  level 
it  is  desirable  to  make  the  length  of  sights  (distance  from 
instrument  to  rod)  as  long  as  practicable,  in  order  to  increase 
the  rate  of  progress.  For  the  best  work  this  distance  may  be 
from  100  to  300  feet,  according  to  the  state  of  the  atmosphere. 
When  the  air  and  ground  differ  greatly  in  temperature  there 
result  innumerable  little  upward  and  downward  currents  of 
air,  the  upward  being  warmer  than  the  downward  currents. 
The  warmer  air  is  more  rarefied  than  the  colder,  and  thus  a 
ray  of  light  passing  from  the  rod  to  the  instrument  passes  alter- 
nately through  denser  and  rarer  media,  each  change  producing 
a slight  refraction  of  the  ray.  This  causes  a peculiar  tremulous 


74 


SC/A*  VE  YING. 


condition  of  the  image  in  the  telescope,  so  that  it  is  difficult  to 
determine  just  vvliat  part  of  it  is  covered  by  the  cross-hairs.  At 
such  times  the  air  is  said  to  be  “ trembling”  or  “ dancing”  or 
“unsteady.”  It  always  occurs  more  or  less  in  clear  weather, 
owing  to  the  earth  then  being  hotter  than  the  air,  and  it  varies 
with  the  quality  of  the  soil,  cinders  or  gravel  being  veiy  bad. 
When  the  air  is  in  this  condition  the  length  of  sights  should 
be  shortened. 

The  back  ajid  fore  sights  for  any  setting  of  the  instrument 
should  always  be  equal  in  length.  Levelling  is  the  only  kind  of 
field-surveying  wherein  the  instrumental  errors  may  be  thor- 
oughly eliminated  without  duplicating  the  observations.  This 
may  be  done  in  levelling  by  making  the  back  and  fore  sights 
of  equal  length.  For,  since  the  difference  between  back  and 
fore  sights  is  always  the  quantity  used,  it  follows  that  if  both 
are  too  large  or  too  small  by  the  same  amountj  the  difference 
will  be  unchanged.  If,  when  the  bubble  is  in  the  middle  of 
its  tube,  the  line  of  sight  is  inclined  upwards  by  a given  small 
angle,  then  it  has  this  relation  to  the  horizontal  on  both  fore 
and  back  sights,  and  if  the  lengths  of  sights  zvere  equal  the  fore 
and  back  rod-readings  were  equally  in  error.  It  is  therefore 
very  desirable  that  these  sights  should  be  made  of  equal  length. 
Moreover,  the  effect  of  the  earth’s  curvature  is  eliminated  by 
so  doing,  however  long  the  sights  may  be.  There  are  other 
kinds  of  errors  that  are  not  eliminated  by  this  means,  but  those 
that  are  eliminated  are  of  sufficient  importance  to  warrant 
great  care  to  secure  equal  sights  for  each  setting.  If  it  is  impos- 
sible to  do  this  at  any  time,  the  inequality  should  be  balanced 
off  at  the  next  one  or  two  settings,  by  making  them  unequal 
in  the  opposite  direction  by  the  same  amount.  The  equality 
of  sights  can  be  determined  by  pacing  with  sufficient  accuracy. 

74.  Bench-marks  are  fixed  points  of  more  or  less  perma- 
nent character  whose  elevations  are  determined  and  recorded 
for  future  reference.  The  general  and  particular  location  of  a 


ADJUSTMENT,  USE,  AND  CAEE  OF  INSTRUMENTS.  75 


bench-mark  should  be  so  distinctly  described  that  any  one 
could  find  it  from  its  description.  Whenever  the  work  is  tem- 
porarily interrupted  a temporary  bench-mark  is  set,  such  as 
a substantial  stake  driven  into  the  ground,  or  a spike  in  the 
root  of  a tree.  The  prime  requisite  of  a good  bench-mark  is 
that  it  shall  not  change  its  elevation  during  the  period  in 
which  it  is  to  be  used.  If  this  period  is  not  more  than  two  or 
three  years,  a spike  driven  in  the  spreading  root  of  a tree 
near  the  trunk  and  well  above  ground  will  serve.  The  wood 
should  be  trimmed  away  from  it  so  as  to  leave  a projecting 
spur  that  will  not  be  overgrown.  The  tree  itself  should  then 
be  marked  by  notching  or  otherwise,  and  carefully  located  in 
the  description. 

If  the  mark  is  to  serve  for  from  five  to  fifty  years,  stone  or 
brick  structures  or  natural  rock  should  be  selected.  The  water- 
tables,  or  corners  of  stone  steps,  of  buildings,  copings  of  founda- 
tion and  retaining  walls,  piers  and  abutments  of  bridges,  or 
copper  bolts  leaded  in  natural  rock  may  serve.  If  artificial 
structures  are  chosen,  those  should  be  selected  which  have 
probably  settled  to  a fixed  position,  and  for  this  reason  old 
structures  are  preferable  to  new  ones. 

When  stakes  are  used  for  temporary  benches  it  is  often 
advisable  to  set  two  or  even  three  for  a check.  In  this  case 
the  mean  elevation  is  the  elevation  used.  In  starting  from 
such  a series  of  benches  there  would  be  as  many  back-sights 
for  the  first  setting  of  the  instrument  as  there  were  benches, 
the  mean  of  which,  added  to  the  mean  elevation  of  the  benches, 
would  give  the  height  of  instrument.  In  running  a continuous 
line  of  levels  it  is  advisable  to  set  a bench-mark  at  least  as 
often  as  one  to  the  mile. 

75.  The  Record  in  differential  levelling  is  very  simple. 
The  bubble  always  being  put  in  the  middle  of  the  tube,  and 
the  rod-positions  chosen  equally  distant  from  the  instrument, 
the  bubble-reading  and  the  length  of  sights  may  be  omitted 


76 


SUR  VE  YING. 


from  the  record,  unless  some  knowledge  of  the  distance  run  is 
desired,  when  the  length  of  sights  may  be  inserted. 

Form  of  Record  for  Differential  Levelling. 


No.  of 
Station. 

Back-sights. 

Fore-sights. 

Elevation  of 
Mean  Benches. 

Remarks. 

3.426 

3.878 

96.301 

B.  S.  on  B.  M.  31 
31a 

1 

2 

3- 652 

4- 517 

4.879 

3.472 

3 

3-216 

4.361 

4.873 

94.718 

F.  S.  on  B.  M.  32 
32  a 

4 617 

+11.385 

— 12.968 

+11.385 

- 1.583 

It  will  be  seen  that  the  mean  of  the  readings  on  the  two 
bench-marks  was  used  in  each  case.  The  back-sights  being 
essentially  positive  and  the  fore-sights  essentially  negative, 
these  signs  are  prefixed  to  the  sums,  and  the  algebraic  sum  of 
these  gives  the  elevation  of  the  forward  above  or  below  the 
rear  benches.  This  added  to  the  elevation  of  the  initial  point 
gives  the  elevation  of  the  final  point.  These  points  are  the 
mean  elevation  of  two  bench-marks  in  the  example  given. 

76.  The  Field-work  should  be  done  with  great  care  if 
the  best  results  are  to  be  obtained.  The  instrument  should  be 
adjusted  every  day,  especially  the  parallelism  of  bubble-axis 
and  line  of  sight.  The  instrument  and  rod  should  both  be  set 
in  firm  ground.  An  iron  pin,  about  one  inch  square  at  top, 
six  to  eight  inches  long,  and  tapering  to  a point,  should  be 
used  for  the  turning-point.  A rope  or  leather  noose  should  be 
passed  through  an  eye  at  top  to  serve  as  a handle.  To  hold 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  7/ 


the  rod  upright  the  rodman  should  stand  squarely  behind  it 
and  keep  it  balanced  on  the  pin.  When  the  target  is  set  and 
clamped  the  rodman  reads  it  and  records  it  on  a paper  he  car- 
ries for  the  purpose.  He  then  carries  it  to  the  observer,  if  it 
was  a back-sight  reading,  or  he  awaits  the  coming  of  the  ob- 
server if  it  was  a fore-sight  reading,  when  the  observer  also 
reads  it  and  records  it  in  his  note-book.  The  rodman  then 
calls  off  his  reading,  and  the  observer  notes  its  agreement  with 
his  recorded  reading.  In  this  way  two  wholly  independent 
readings  are  obtained  and  any  erroneous  reading  corrected. 
Errors  of  one  foot  or  one  tenth  are  not  very  uncommon  in 
reading  target  rods.  The  rodman  should  be  especially  careful 
to  protect  the  turning-point  from  all  disturbances  between  the 
forward  and  back  readings  upon  it.  The  observer  must  not 
only  obtain  an  accurate  bisection  on  the  target,  but  he  must 
know  that  the  bubble  is  accurately  in  the  centre  of  the  tube 
when  this  bisection  is  obtained.  When  the  observer  walks  for- 
ward to  set  his  instrument  he  counts  his  paces,  and  takes  as 
long  a sight  as  the  nature  of  the  ground  or  the  condition  of 
the  atmosphere  will  allow.  When  the  rodman  comes  up  he 
counts  his  paces  to  the  instrument  and  then  goes  the  same  dis- 
tance in  advance.  Thus  the  observer  controls  the  lenccth  of 
sights,  making  them  whatever  he  likes ; and  it  is  the  business 
of  the  rodman  to  see  that  the  back-  and  fore-sight  for  every 
instrument-station  are  equal. 

PROFILE  LEVELLING. 

77.  In  Profile  Levelling  the  object  is  to  obtain  a profile  of 
the  surface  of  the  ground  on  certain  established  lines.  Here 
both  the  distances  from,  and  the  elevation  above,  some  fixed 
initial  point  must  be  obtained.  When  the  line  is  laid  out 
stakes  are  usually  driven  every  hundred  feet,  these  positions 
being  obtained  by  a chain  or  tape.  It  is  now  the  business  of 
the  leveller  to  obtain  the  elevation  of  the  ground  at  each  of 


78 


SURVEYING. 


these  stakes,  and  at  as  many  other  intermediate  points  as  may 
be  necessary  to  enable  him  to  draw  a fairly  accurate  profile  of 
the  ground.  The  lOO-foot  stakes  are  usually  numbered,  and 
these  numbers  are  entered  on  the  level  record.  The  inter- 
mediate points  are  called  pluses.  Thus,  a point  40  feet  beyond 
the  twenty-fifth  lOO-foot  stake  is  called  25  40,  being  really 

2540  feet  from  the  initial  point.  It  is  evident  that  no  plus- 
distance  can  be  more  than  loo  feet,  and  these  are  usually  paced 
by  the  rodman.  The  intermediate  points  are  selected  with 
reference  to  their  value  in  determining  the  profile.  These  are 
points  where  the  slope  changes,  being  mostly  maximum  and 
minimum  points,  or  the  tops  of  ridges  and  bottoms  of  hollows. 
Turning-points  are  selected  at  proper  distances,  depending  on 
the  accuracy  required,  and  these  may  or  may  not  be  points  in 
the  line  whose  profile  is  desired.  The  levelling-instrument  also 
is  not  set  on  line,  if  it  is  found  more  convenient  to  set  it  off  the 
line. 

In  profile  levelling,  since  absolute  elevations  with  reference 
to  the  datum-plane  are  to  be  obtained  from  every  instrument- 
position,  it  is  necessary  to  find  the  height  of  instrument  above 
datum  for  every  setting,  and  from  this  height  of  instrument, 
obtained  by  a single  back-sight  reading  on  the  last  turning- 
point,  the  elevations  of  any  number  of  points  are  found  by  sub- 
tracting the  readings  upon  them. 

78.  The  Record  in  profile  levelling  is  much  more  elaborate 
than  in  differential  levelling.  The  following  form  is  considered 
very  convenient  for  profile-work  where  the  line  has  been  laid 
out  and  lOO-foot  stakes  set  :* 


* This  sample  page  was  contributed  \.o  Engineering  News  in  June,  1879,  and 
the  form  of  record  is  credited  to  Mr.  E.  S.  Walters,  a railroad  engineer  of  large 
experience. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  79 


Guatemala  and  Honolulu  Railroad.  Feb.  30,  1876, 


B.  S. 

El.  of  T.  P. 
and  B.  M. 

F.  S. 

H.  I. 

I.  S. 

S.  E. 

Sta. 

Remarks. 

10.552 

195-497 

206.049 

B.  M. 
188  + 44 

189 

190 

+ 30 

191 
-f-  20 

192 

T.  P. 

193 
+ 50 

B.  M. 

194 

195 

T.  P. 

196 

197 
4-60 

+ ^5 

198 

199 

T.  P. 

+ 35 

200 

201 



9-32 
II  .41 

7.01 
2.07 
1.62 
0.38 
0.  82 

196.73 
194.64 
199.04 
203.98 
204.43 
205.67 
205 . 13 

0-515 

202.797 

3-252 

203.312 

3. 10 

2.70 

5-264 

8.20 

9-35 

200.21 

200.61 

198.048 

195. II 
193.96 

3.411 

194.840 

8.472 

198.251 

4.28 

5.06 

7.20 

10,60 

7.00 

5.46 

193-97 

193-19 

191-05 

187.65 

191.25 

192.79 

9-527 

195.083  1 3.168 

204.610 



10.25 

8.62 

6.04 

194.36 

195-99 

198-57 

24.005 

14.892 

14.892 

1 

9-”3 

In  the  above  headings,  B.  S.  denotes  back-sight;  F.  S.,  fore-sight;  I.  S., 
intermediate  sight  ; H.  I.,  height  of  instrument  ; T.  P.,  turning-point  ; B.  M., 
bench-mark  ; S.  E.,  surface-elevation  ; Sta.,  station. 


It  will  be  noted  that  there  is  but  one  back-sight  and  one 
height  of  instrument  for  each  setting.  The  back-sight  and 
fore-sight  readings  from  the  same  instrument-station  are  not 
found  here  on  the  same  line,  as  in  differential  levelling,  but  the 
fore-  and  back-readings  on  the  same  turning-point  are  on  the 
same  line.  Thus,  the  rod  was  first  read  on  the  bench-mark 
whose  elevation  was  known  to  be  195.497  feet  above  datum. 
The  reading  on  this  bench  was  10.552,  thus  giving  a height  of 


8o 


SUR  VE  Y I EG. 


instrument  of  206.049.  This  is  marked  B.  M.  in  the  station 
column,  and  evidently  has  but  one  reading  upon  it  in  starting 
the  work  from  it.  A series  of  intermediate  sights  are  then 
taken  at  various  lOO-foot  stakes  and  pluses,  the  readings  on 
which,  when  subtracted  from  the  H.  I.,  give  the  surface-eleva- 
tions at  those  points.  When  the  work  has  progressed  as  far  in 
front  of  the  instrument  as  the  B.  M.  was  back  of  it,  a turning- 
point  is  set,  and  the  reading  upon  it  recorded  in  the  column  of 
fore-sights.  This  reading  was  3.252,  which,  subtracted  from  the 
H.  I.  206.049,  gives  202.797  as  the  elevation  of  the  turning- 
point.  The  instrument  is  now  moved  forward  and  a back- 
sight reading  taken  upon  this  T.  P.  of  0.515,  which  added  to 
202.797  gives  203.312  as  the  new  H.  I.  At  this  setting  a new 
bench  was  established  by  taking  an  intermediate  sight  upon  it 
of  5.264,  and  writing  the  elevation  in  the  B.  M.  column  instead 
of  in  the  S.  E.  column.  The  readings  on  bench-marks  and 
turning-points  are  made  to  thousandths,  while  the  intermediate 
sights  for  surface-elevation  are  read  only  to  hundredths  of  a 
foot.  The  last  height  of  instrument  is  checked  by  adding  the 
back-sights  and  fore-sights,  taking  the  difference  and  applying 
it  to  the  elevation  of  the  initial  point  with  its  proper  sign,  re- 
membering that  back-sights  are  positive  and  fore-sights  negative. 
The  profile  is  now  constructed  by  the  data  found  in  the  S.  E. 
and  Sta.  columns,  these  being  adjacent  to  each  other.  One 
of  the  great  merits  of  this  form  of  record  is  that  wherever 
it  is  necessary  to  combine  any  two  numbers  by  addition  or 
subtraction,  they  are  found  in  adjacent  columns.  In  construct- 
ing the  profile,  some  kind  of  profile  or  cross-section  paper  is 
used,  and  the  horizontal  scale  made  much  smaller  than  the 
vertical.  Thus,  if  the  horizontal  scale  were  400  feet  to  the 
inch,  the  vertical  scale  might  be  10  or  20  feet  to  the  inch. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  8 1 


LEVELLING  FOR  FIXING  A GRADE. 

79.  In  fixing  a grade  the  profile  may  be  obtained  and 
the  grade  marked  upon  it.  The  vertical  distance  between  the 
surface-line  and  the  grade-line,  at  any  point  is  the  depth  of 
cut  or  fill  at  that  point,  and  this  may  be  marked  on  the  line 
stakes  at  once,  without  the  aid  of  the  level  or  rod,  if  only  the 
centre  depths  are  desired,  as  in  the  case  of  a ditch  or  trench. 
If  the  sides  are  to  have  a required  slope,  however,  the  level 
and  rod  are  necessary  to  fix  the  horizontal  distance  of  the 
limiting  or  “ slope”  stakes  from  the  centre  stakes  whenever 
the  ground  is  not  strictly  a level  surface.  This  operation  is 
called  “ cross-sectioning,”  and  is  described  in  Chapter  XIII., 
on  Determination  of  Volumes. 

If  the  grade  be  known  before  the  profile  is  determined,  to- 
gether with  the  absolute  elevation  of  the  initial  point,  as  is 
sometimes  the  case  with  ditches  and  trenches  for  pipe  lines  or 
sewers,  then  the  depth  of  cut  (or  fill)  may  be  at  once  deter- 
mined and  marked  on  the  line  stakes  when  the  profile  is  taken. 
The  form  of  record  might  be  the  same  as  given  above  for  pro- 
file levelling,  with  the  addition  of  two  columns  after  the  “ Sta- 
tion” column,  one  being  Elevation  of  Grade,  and  the  other  Cut 
or  Fill.  The  elevation  of  grade  would  be  found  for  each  pro- 
file point  by  adding  if  an  up,  and  subtracting  if  a down,  grade, 
the  differences  of  elevation  corresponding  to  the  successive 
distances  in  the  profile.  The  difference  between  the  corre- 
sponding “surface-elevation”  and  “ elevation  of  grade”  would 
be  the  cut  or  fill  at  each  point,  which  could  be  at  once  taken 
out  and  marked  on  the  line  stake. 

THE  HAND-LEVEL. 

80.  Locke’s  Hand-level  is  a very  convenient  little  instru- 
ment for  rough  work,  such  as  is  done  on  reconnaissance  expedi- 
tions. It  consists  of  a telescope  with  a bubble  attached  in 

6 


82 


SURVEYING. 


such  a way  that  the  position  of  the  bubble  is  seen  by  looking 
through  the  telescope.  A horizontal  line  of  sight  is  thus 
readily  determined.  It  is  supposed  to  be  adjusted  once  for  all. 


Fig.  i6. 


EXERCISES  WITH  THE  LEVEL. 

81.  Adjust  the  bubble  to  the  line  of  sip^ht  by  the  first,  or  indirect,  method, 
and  then  test  it  by  the  second,  or  direct,  method.  If  this  second  method  does 
not  show  it  to  be  in  adjustment,  where  does  the  error  lie  ? 

82.  Cause  the  line  of  sight  and  bubble-axis  to  make  a considerable  angle 
with  each  other  (that  is,  put  it  badly  out  of  adjustment  in  this  particular),  and 
level  around  a block  or  two,  closing  on  the  starting-point,  being  careful  to 
make  back  and  fore  sights  as  nearly  equal  as  possible.  Of  course  the  final 
elevation  of  the  point  should  agree  with  the  assumed  initial  elevation.  The 
difference  of  these  elevations  is  the  error  of  closure  of  the  level  polygon.  If 
the  back  and  fore  sights  were  exactly  equal  this  should  be  zero,  notwithstand- 
ing the  erroneous  adjustment. 

83.  Put  the  instrument  in  accurate  adjustment,  and  level  over  the  same 
polygon  as  before,  making  the  back  and  fore  sights  quite  unequal,  and  note  the 
error  of  closure.  If  the  instrument  were  in  exact  adjustment  and  there  were 
no  errors  of  observation,  should  the  error  of  closure  be  zero  ? 

84.  Range  out  a line  on  uneven  ground  about  a half-mile  in  length,  and  set 
stakes  every  hundred  feet.  Let  each  student  determine  the  profile  indepen- 
dently. When  all  have  finished,  let  them  copy  their  profiles  on  the  same  piece 
of  tracing-cloth,  starting  at  a common  point.  The  vertical  scale  should  be 
large,  so  as  to  scatter  the  several  profile  lines  sufficiently  on  the  tracing.  Each 
profile  should  be  in  a different  color  or  character  of  line. 

85.  Select  a line  on  nearly  level  ground,  about  a half-mile  in  length.  Estab- 
lish a substantial  bench-mark  at  each  end.  Let  each  student  determine  the 
difference  of  elevation  of  these  benches  twice,  running  forward  and  back.  See 
if  the  results  are  affected  by  the  direction  in  which  the  line  is  run. 

If  each  student  could  do  this  several  times  some  evidence  would  be  ob- 
tained as  to  there  being  such  a thing  as  “ personal  equation”  in  levelling  ; that 
is,  each  person  tending  to  always  obtain  results  too  high  or  too  low.  Why  is 
it  improbable  that  there  could  be  any  personal  equation  in  levelling? 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS,  83 


/ 


CHAPTER  IV. 

INSTRUMENTS  FOR  MEASURING  ANGLES. 

THE  TRANSIT. 

86.  The  Engineer’s  Transit  is  the  most  useful  and 
universal  of  all  surveying-instruments.  Besides  measuring 
horizontal  and  vertical  angles  it  will  read  distances  by  means 
of  stadia  wires,  determine  bearings  by  means  of  the  magnetic 
needle,  do  the  work  of  a solar  compass  by  means  of  a special 
attachment,  and  do  levelling  by  means  of  a bubble  attached 
to  the  telescope.  It  is  therefore  competent  to  perform  all  the 
kinds  of  service  rendered  by  any  of  the  instruments  heretofore 
described,  and  is  sometimes  called  the  “universal  instrument.” 
A cut  of  this  instrument  is  shown  in  Fig.  17.  Fig.  18  is  a 
sectional  view  through  the  axis  of  a transit  of  different 
manufacture. 

The  telescope,  needle-circle,  and  vernier  plates  are  rigidly 
attached  to  the  inner  spindle  which  turns  in  the  socket  Ci 
Fig.  18.  This  portion  of  the  instrument  is  called  the  alidade, 
as  it  is  the  part  to  which  the  line  of  sight  is  attached.  The 
socket  C carries  the  horizontal  limb,  shown  at  B,  and  may 
itself  revolve  in  the  outer  socket  attached  to  the  levelling-head. 
Either  or  both  of  these  connections  may  be  made  rigid  by 
means  of  proper  clamping  devices.  If  the  horizontal  limb  B 
be  clamped  rigidly  to  the  levelling-head  and  the  alidade  spindle 
be  allowed  to  revolve,  then  horizontal  angles  may  be  read  by 
noting  the  vernier-readings  on  the  fixed  horizontal  limb  for 
the  different  pointings  of  telescope.  If  the  horizontal  limb 
itself  be  set  and  clamped  so  that  one  of  the  verniers  reads  zero 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  8$ 


when  the  telescope  is  on  the  meridian,  then  for  any  other 
pointing  of  the  telescope  the  reading  of  this  same  vernier 
gives  the  true  azimuth  of  the  line.  It  is  necessary,  therefore, 
to  have  two  independent  movements  of  telescope  and  horizon- 
tal limb  on  the  same  vertical  axis.  The  magnetic  needle  is 
shown  at  N.  The  plumb-line  is  attached  at  P',  this  should 
always  be  in  the  vertical  line  passing  through  the  centre  of 
the  graduated  horizontal  circle.  This  will  be  the  case  when 


it  is  attached  directly  to  the  axis  itself,  for  this  must  always 
be  made  vertical. 

The  limb  is  graduated  from  zero  to  360°,  and  sometimes 
with  a second  set  of  figures  to  90°  or  180°.  There  are  two 
verniers  reading  on  the  horizontal  limb  180°  apart.  Both  the 
instruments  shown  in  Figs.  17  and  18  have  shifting  centres, 
enabling  the  final  adjustment  of  the  instrument  over  a point 
to  be  made  by  moving  it  on  the  tripod-head.  The  telescope 
is  shorter  than  those  used  in  levelling-instruments  in  order 
that  it  may  be  revolved  on  its  horizontal  axis  without  having 
the  standards  too  high.  It  is  called  a transit  instrument  on 


86 


SUR  VE  YING. 


account  of  this  movement,  which  is  similar  to  that  of  an 
astronomical  transit  used  for  observing  the  passage  (transit) 
of  stars  across  any  portion  of  the  celestial  meridian.  When 
the  telescope  is  too  long  to  be  revolved  in  this  way  the  instru- 
ment is  called  a theodolite.  This  is  the  only  essential  differ- 
ence between  them.*  The  “ plain  transit  ” has  neither  a 
vertical  circle  nor  a bubble  attached  to  the  telescope. 

ADJUSTMENTS  OF  THE  TRANSIT. 

87.  The  Adjustments  of  the  Engineer’s  Transit  are 
such  as  to  cause  (i)  the  instrument  to  revolve  in  a horizontal 
plane  about  a vertical  axis,  (2)  the  line  of  collimation  to  gen- 
erate a vertical  plane  through  the  instrument-axis  when  the 
telescope  is  revolved  on  its  horizontal  axis,  (3)  the  axis  of  the 
telescope-bubble  to  be  parallel  to  the  line  of  collimation,  thus 
enabling  the  instrument  to  do  levelling,  and  (4)*the  vernier  on 
the  vertical  circle  so  adjusted  that  its  readings  shall  be  the 
true  altitude  of  the  line  of  collimation.  These  four  results  are 
attained  by  making  the  following  five  adjustments  : 

88.  First.  To  make  the  Plane  of  the  Plate-bubbles 
perpendicular  to  the  Vertical  Axis. — This  adjustment  is 
the  same  as  with  the  compass.  (One  of  the  plate-bubbles  is 
usually  set  on  one  pair  of  standards.)  Bring  both  bubbles  to 
the  centre,  revolve  180°,  correct  one  half  the  movement  on 
the  levelling-screws  and  the  other  half  by  raising  or  lowering 
the  adjustable  end  of  the  bubble-tube.  Each  bubble  should 
be  brought  parallel  to  a set  of  opposite  levelling-screws  in 
making  this  adjustment,  so  that  the  correcting  for  one  bubble 
does  not  throw  the  other  out.  When  either  bubble  will  main- 
tain a fixed  position  in  its  tube  as  the  instrument  is  revolved 
horizontally,  the  axis  of  revolution  is  vertical.  One  bubble  is 

*The  first  engineer’s  transit  instrument  was  made  by  Wm.  J.  Young  (now 
Young  & Sons),  Pi)iladelphia,  1831.  All  American  engineer’s  altitude-azimuth 
instruments  are  now  made  to  revolve  in  this  way. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  87 


therefore  sufficient  for  making  this  axis  vertical,  but  two  are 
somewhat  more  convenient,  especially  for  indicating  when  the 
axis  has  become  inclined  from  unequal  settling  or  expansion 
while  in  use. 

89.  Second.  To  make  the  Line  of  Collimation  perpen- 
dicular to  the  Horizontal  Axis  of  the  Telescope. — When 
this  is  done,  the  line  of  collimation  will  generate  a plane  when 
the  telescope  is  revolved  on  its  horizontal  axis.  If  the  line  of 
collimation  is  not  perpendicular  to  the  horizontal  axis,  it  gen- 
erates the  surface  of  a cone  when  the  telescope  is  revolved,  the 
axis  of  the  cone  being  the  axis  of  revolution,  and  the  apex 
being  at  the  intersection  of  the  line  of  collimation  with  this 
axis. 

Set  the  instrument  on  nearly  level  ground,  where  a view 
can  be  had  in  opposite  directions.  Set  the  line  of  sight  on  a 
definite  point  a few  hundred  feet  away.  Revolve  the  telescope 
and  set  another  point  in  the  opposite  direction.  Revolve  the 
alidade  until  the  line  of  sight  comes  upon  the  first  point.  Re- 
volve the  telescope  again  and  fix  a third  point  on  the  line  of 
sight  beside  the  second  point  set.  Measure  off  one-fourth  the 
distance  between  these  two  points  from  the  last  point  set,  and 
bring  the  line  of  sight  to  this  position  by  moving  the  reticule 
laterally.  This  movement  of  the  reticule  is  direct  in  an  erect- 
ing instrument  and  reversed  in  an  inverting  instrument. 

The  student  should  illustrate  the  correctness  of  this  method 
by  means  of  a figure.  The  four  pointings  were  the  intersec- 
tions of  a diametral  horizontal  plane  with  the  surfaces  of  the 
the  two  cones  generated.  These  cones  were  pointed  in  oppo- 
site directions,  but  had  one  element  in  common,  being  the  two 
pointings  to  the  first  point.  The  two  opposite  elements 
diverged  by  four  times  the  difference  between  the  semi-angle 
of  the  cone  (subtended  by  the  line  of  collimation  and  the  axis 
of  rotation)  and  90°. 

90.  Third.  To  make  the  Horizontal  Axis  of  the  Tele- 


88 


SURVEYING. 


scope  perpendicular  to  the  Axis  of  the  Instrument. — When 
this  is  done  the  former  is  horizontal  when  the  latter  is  vertical, 
and,  the  second  adjustment  having  been  made,  the  line  of  sight 
will  generate  a vertical  plane  when  the  telescope  is  revolved. 

Set  the  instrument  firmly  and  level  it  carefully.  Suspend 
a plumb-line  some  20  or  30  feet  long,  some  15  or  20  feet  from 
the  instrument.  The  weight  should  rest  in  a pail  of  water  and 
the  string  should  be  hung  from  a rigid  support.  There  should 
be  no  wind,  and  the  cord  should  be  small  and  smooth.  A small 
fish-line  is  very  good.  Care  must  be  exercised  that  the  weight 
does  not  touch  the  bottom  of  the  pail  from  the  stretching  of 
the  cord.  Set  the  line  of  sight  carefully  on  the  cord  at  top, 
the  plate-bubbles  indicating  a strictly  vertical  instrument-axis. 
Clamp  both  horizontal  motions  and  bring  the  telescope  to  read 
on  the  bottom  portion  of  the  cord.  The  cord  is  apt  to  swing 
to  and  fro  slightly,  but  its  mean  position  can  be  chosen.  If  the 
line  of  sight  does  not  correspond  to  this  mean  position,  raise 
or  lower  the  adjustable  end  of  the  horizontal  axis  until  this 
test  shows  the  line  of  sight  to  revolve  in  a vertical  plane. 
Constant  attention  must  be  given  to  the  plate-bubbles  to  see 
that  they  do  not  indicate  an  inclined  vertical  axis. 

Or,  two  points  nearly  in  a vertical  line  may  be  used,  as  the 
top  and  bottom  of  the  vertical  corner  of  a building.  Set  on 
the  top  point  and  revolve  to  the  bottom  point.  Note  the 
relation  of  the  line  of  sight  to  this  point.  Revolve  180°  about 
both  vertical  and  horizontal  axes,  and  set  again  on  the  top 
point.  Lower  the  telescope  again  and  read  on  the  bottom 
point.  If  the  telescope-axis  of  revolution  is  horizontal,  the 
second  pointing  at  bottom  should  coincide  with  the  first.  If 
not,  adjust  for  one  half  the  difference  between  these  two 
bottom  readings. 

It  will  be  noted  that  the  second  and  third  adjustments  are 
necessary  to  the  accomplishment  of  the  second  result  cited  in 
art.  87. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  89 


91.  Fourth.  To  make  the  Axis  of  the  Telescope-bub- 
ble parallel  to  the  Line  of  Collimation. — This  adjustment 
is  performed  by  means  of  the  “peg-adjustment/’  as  described 
in  art.  65,  p.  65,  second  method.  The  height  of  the  instrument 
may  now  be  measured  to  the  centre  of  the  horizontal  axis  if  it 
be  found  more  convenient  than  sighting  backwards  through 
the  telescope.  When  this  adjustment  is  made  the  instrument 
is  competent  to  do  levelling  the  same  as  the  levelling-instru- 
ment. The  telescope  is  not  quite  so  stable,  however,  in  the 
transit  because  it  is  mounted  on  an  axis  instead  of  in  two  rigid 
wyes. 

92.  Fifth.  To  make  the  Vernier  of  the  Vertical  Circle 
read  Zero  when  the  Line  of  Sight  is  Horizontal. — Having 
made  the  axis  of  the  telescope-bubble  parallel  to  the  line  of 
sight,  bring  this  into  the  centre  of  its  tube,  and  adjust  the 
vernier  of  the  vertical  circle  till  it  reads  zero  on  the  limb.  If 
this  vernier  is  not  adjustable,  the  reading  in  this  position  is  its 
index  error.  The  line  of  sight  might  still  be  adjusted  to  the 
vernier  by  moving  the  reticule,  and  then  adjusting  the  bubble 
to  the  line  of  sight.  To  do  this  use  the  “peg-adjustment”  as 
described  in  art.  65,  making  the  vertical  circle  read  zero  each 
time,  and  paying  no  attention  to  the  telescope-bubble.  Correct 
the  line  of  sight  by  \ d.  as  given  by  Eq.  (2),  p.  66,  by  moving 
the  reticule,  and  this  should  give  a horizontal  pointing  for  a 
zero-reading  of  the  vertical  circle.  Then  adjust  the  bubble  to 
this  reading  by  bringing  it  to  the  centre  of  the  tube  by  means 
of  the  vertical  motion  at  one  end  of  the  bubble-tube.  If  the 
reticule  is  disturbed  after  making  the  second  adjustment,  that 
adjustment  should  be  tested  again  to  see  if  it  had  been  dis- 
turbed. 

93.  Relative  Importance  of  the  Adjustments. — The  first 
adjustment  is  important  in  all  horizontal  and  vertical  angular 
measurements.  In  measuring  vertical  angles  the  error  may  be 
the  full  amount  of  the  deviation  of  the  vertical  axis  from  the 


90 


SUR  VE  YING. 


vertical,  and  in  measuring  horizontal  angles  something  very 
much  less  than  this. 

The  second  adjustment  is  more  important  in  the  running  of 
a straight  line  by  revolving  the  telescope  than  in  any  other  kind 
of  work,  for  here  the  error  in  the  continuation  of  the  line  is 
twice  the  error  of  adjustment.  It  is  also  important  in  measur- 
ing horizontal  angles  between  points  not  in  the  same  horizontal 
plane. 

The  third  adjustment  is  most  important  in  the  measure- 
ment of  horizontal  angles  between  points  not  in  the  same  hori- 
zontal plane,  as  in  the  determination  of  the  azimuth  of  a line 
by  an  observation  on  a circumpolar  star. 

The  fourth  and  fifth  adjustments  are  important  only  in 
levelling  operations,  either  by  reading  the  vertical  angle  or  by 
the  use  of  the  bubble. 

INSTRUMENTAL  CONDITIONS  AFFECTING  THE  ACCURATE 
MEASUREMENT  OF  HORIZONTAL  ANGLES.* 

94.  Eccentricity. — This  is  of  two  kinds:  (i)  eccentricity  of 
centres,  and  (2)  eccentricity  of  verniers.  If  the  axis  of  the  coni- 
cal outer  socket  C,  Fig.  18,  is  not  exactly  in  the  centre  of  the 
graduated  limb  B,  then  when  the  telescope  with  the  vernier 
plates  V are  revolved  in  this  socket,  the  verniers  will  have  an 
eccentric  motion  with  reference  to  the  graduated  limb.  If  the 
line  joining  the  zeros  of  the  verniers  passes  through  the  axis 
of  the  socket,  it  is  evident  that  there  is  but  one  position  of 
these  verniers  which  will  give  readings  on  the  limb  180°  apart, 
and  that  is  when  both  centres  lie  in  this  diametral  line.  For 
all  other  positions  of  the  verniers,  one  of  them  will  read  as 
much  too  large  as  the  other  does  too  small ; so  that  if  the  mean 

* For  extended  discussions  of  this  subject,  see  Bauernfeind’s  “ Vermessunjrs- 
kunde,”  ^ 144.  vol,  i.,  and  Jordan’s  “ Handbuch  der  Vermessungskunde,”  § 88. 
vol,  i.  Also  translations  from  these,  by  Prof.  Eisenmann,  in  Journal  of  the 
Association  of  Engineering  Societies,  vol  iv.  p.  ig6. 


ADJUSTMENT,  USE,  AND  CATE  OF  INSTRUMENTS.  9 1 


of  the  two  vernier-readings  be  taken,  this  error  from  eccentric- 
ity would  be  eliminated. 

Eccentricity  of  verniers  is  due  to  their  zeros  not  falling  on 
a diametral  line  through  the  axis  of  the  spindle ; in  other 
words,  they  are  not  i8o°  apart.  This  involves  no  error  in 
measuring  horizontal  angles.  It  is  convenient,  however,  to 
have  the  verniers  read  exactly  i8o°  apart.  In  any  case,  read- 
ing of  both  verniers  and  taking  the  mean  eliminates  all  errors 
from  eccentricity.  An  eccentricity  of  centres  of  one  one-tJiou- 
sandth  of  an  inch  would  cause  a maximum  error  of  T-o8"  on  a 
six-inch  circle  if  but  one  vernier  were  read.  It  is  not  unusual 
for  an  instrument  to  have  an  eccentricity  of  centres  of  several 
times  this  amount,  either  from  wear  or  from  faulty  construction, 
or  both.  The  necessity  for  reading  both  verniers  in  all  good 
work  is  therefore  apparent. 

95.  Inclination  of  Vertical  Axis. — The  horizontal  angle 
between  points  at  different  elevations  is  obtained  by  measuring 
the  horizontal  angle  subtended  by  two  vertical  planes  passing 
through  these  points  and  the  point  of  observation.  These 
vertical  planes  are  the  planes  described  by  the  line  of  sight  as 
the  telescope  is  revolved.  By  this  means  the  points  may  be 
said  to  be  projected  vertically  on  the  horizontal  plane  and 
then  the  angle  measured.  If  the  vertical  axis  of  the  instru- 
ment is  somewhat  inclined,  these  projecting  planes  are  not  ver- 
tical, neither  do  they  have  the  same  inclination  to  the  horizon 
on  different  parts  of  the  limb.  The  projecting  planes  through 
two  points  will  therefore  neither  be  vertical  nor  equally  in- 
clined to  the  horizon.  The  measured  horizontal  angle  thus 
obtained  will  therefore  be  in  error.  The  vertical  axis  is  always 
inclined  when  the  plate-bubbles  are  not  in  adjustment  or  when 
they  do  not  show  a level  position. 

If  the  axis  be  inclined  5'  from  the  vortical,  and  leadings  be 
taken  on  points  60°  apart,  one  being  10°  above  and  the  other 
10°  below  the  horizon,  the  maximun  error  from  this  source 


92 


SUR  VE  YING. 


would  be  about  i'.  If  the  inclination  in  this  case  were  i°,  the 
maximum  error  would  be  i8'.  This  shows  the  importance  of 
keeping  the  plate-levels  in  adjustment  and  of  watching  them 
during  the  progress  of  the  work  to  see  that  they  remain  in  the 
centre. 

96.  Inclination  of  Horizontal  Axis  of  Telescope. — 

This  causes  the  plane  generated  by  the  line  of  sight  to  be  in- 
clined from  the  vertical  as  much  as  the  axis  of  revolution  is 
from  the  horizontal.  The  projecting  planes  are  therefore  all 
equally  inclined,  and  the  resulting  error  in  horizontal  angle  is 
a function  of  the  difference  of  elevation  of  the  two  points.  If 
one  point  is  10°  above  and  the  other  10°  below  the  horizon, 
and  if  the  inclination  of  the  axis  is  5',  the  resulting  error  in 
the  measurement  of  the  horizontal  angle  is  T-45".  This  error 
is  not  a function  of  the  size  of  the  horizontal  angle,  and  would 
be  the  same  for  two  points  in  the  same  vertical  plane,  the  in- 
strument indicating  a horizontal  angle  of  T 45"  between  them 
for  the  case  here  chosen.  In  making  the  adjustment  of  the 
horizontal  axis  by  means  of  the  plumb-line,  if  the  line  be  15 
feet  distant  and  suspended  15  feet  above  the  instrument,  then 
the  pointing  to  the  top  will  have  an  altitude  of  45°.  In  this 
case  the  angular  error  made  in  bisecting  the  plumb-line  will  be 
the  angular  divergence  of  the  axis  of, rotation  from  the  hori- 
zontal. If  the  combined  error  of  the  two  bisections  be  o.  05  in., 
the  angular  error  in  the  adjustment  will  be  i'.  The  adjust- 
ment may  readily  be  made  closer  than  this. 

Errors  from  this  source  are  eliminated  by  revolving  the 
telescope  and  reading  the  same  angle  in  the  reversed  position. 
The  mean  of  the  two  values  will  be  independent  of  this  error. 
If  many  measurements  are  made  of  one  angle,  there  should  be 
an  equal  number  with  telescope  direct  and  reversed. 


The  student  should  show  by  a figure  how  this  elimination  is  effected  by  the 
reversal  of  the  telescope. 


/ 


ADJUSTMENT,  USE,  AND  CATE  OF  INSTRUMENTS.  93 


97.  The  Line  of  Collimation  not  being  Perpendicular 
to  the  Horizontal  Axis. — This  causes  the  projecting  planes 
to  be  conical  surfaces,  which  become  vertical  on  the  horizon. 
Since  the  error  of  collimation  is  necessarily  a small  angle,  thus 
causing  the  conical  surface  to  be  very  nearly  a plane,  and  since 
this  surface  is  vertical  on  the  horizon,  the  resulting  error  in 
measuring  horizontal  angles  is  very  small  unless  the  difference 
in  the  elevations  of  the  points  is  very  great.  If  the  points  are 
distant,  as  they  always  are  in  the  accurate  measurement  of 
horizontal  angles,  then  their  angular  elevation  is  necessarily 
small,  so  that  this  source  of  error  is  insignificant  in  this  kind 
of  work.  When  straight  lines  are  prolonged  by  reversing  the 
telescope,  however,  this  adjustment  becomes  very  important, 
for  the  error  then  enters  the  work  with  twice  its  angular 
amount.  It  is  eliminated  by  revolving  the  alidade  until  the 
line  of  collimation,  with  telescope  reversed,  falls  again  on  the 
rear  point,  and  again  revolving  the  telescope.  The  point  now 
falls  as  far  on  one  side  of  the  true  position  as  it  before  did  on 
the  other.  The  middle  point  lies  therefore  in  the  line  pro- 
longed. 

Let  the  student  illustrate  by  diagram. 

THE  USE  OF  THE  TRANSIT. 

98.  To  measure  a Horizontal  Angle. — Having  centred 
the  instrument  over  the  vertex  of  the  angle  required,  take  a 
pointing  to  one  of  the  points  and  clamp  both  alidade  and 
limb.  Make  the  final  bisection  by  means  of  either  tangent- 
screw.  Read  the  two  verniers,  and  record  them,  calling  one 
the  reading  of  vernier  A and  the  other  of  vernier  B.  Loosen 
the  alidade  clamp  and  turn  upon  the  second  point,  clamp,  and 
set  by  the  upper  tangent-screw.  Read  both  verniers  again. 
Correct  the  readings  of  vernier  A by  half  the  difference  be- 
tween the  A and  B readings  in  each  case.  The  difference 
between  these  corrected  readings  is  the  value  of  the  angle. 


94 


SURVEYING. 


I^e  careful  not  to  disturb  the  lower  clamp- or  tangent-screw 
after  reading  on  the  first  point.  If  there  are  two  abutting 
tangent-screws  for  the  lower  plate,  be  sure  that  both  arc 
snug,  otherwise  there  may  be  some  play  here  which  would 
allow  the  limb  to  shift  its  position,  in  which  case  the  true  angle 
would  not  be  obtained.  If  there  is  but  a single  tangent-screw 
working  against  a spring  on  the  other  side  of  the  armature, 
as  shown  in  Fig.  i6,  then  there  can  be  no  lost  motion  unless 
the  friction  on  the  axis  is  greater  than  the  spring  can  over- 
come, which  should  never  be  the  case. 

Do  not  set  the  clamp-screws  too  tightly,  as  it  strains  and 
wears  out  the  instrument  unnecessarily.  A very  gentle  press- 
ure is.  usually  sufficient  to  prevent  slipping.  This  caution 
applies  equally  well  to  all  levelling-,  adjusting-,  and  connecting- 
screws  in  the  instrument.  The  young  observer  is  generally 
inclined  to  set  them  up  hard,  as  he  would  in  heavy  iron-work. 
It  must  be  remembered  that  brass  is  a soft  material,  easily  dis- 
torted and  worn,  and  that  the  parts  should  be  strained  as  little 
as  possible  to  insure  against  movement  in  ordinary  handling. 

The  subject  of  measurement  of  horizontal  angles  is  further 
discussed  in  Chapter  XIV.,  on  Geodetic  Surveying. 

99.  To  measure  a Vertical  Angle. — Vertical  angles  are 
usually  referred  to  the  horizon,  and  are  angles  of  elevation  or 
depression  above  that  plane.  If  the  vernier  on  the  vertical 
circle  has  been  properly  adjusted  (or  its  index  error  determined 
in  case  it  is  not  adjustable  and  the  line  of  sight  has  not  been 
adjusted  to  it),  then  the  altitude  of  a point  is  obtained  at  once 
by  turning  the  line  of  sight  upon  it  and  reading  the  vertical 
angle.  Special  attention  must  here  be  given  to  the  bubble 
parallel  to  the  vertical  circle,  for  it  is  on  this  bubble  that  the 
accuracy  of  the  result  wholly  depends.  If  there  is  but  one 
vernier,  it  is  designed  to  read  both  ways,  as  is  shown  in  Figs.  5 
or  6,  p.  19.  In  this  case  errors  of  eccentricity  cannot  be  elim- 
inated. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  Q5 


To  eliminate  errors  of  adjustment  of  the  plate-bubbles  and 
of  the  vernier  on  the  vertical  circle,  revolve  the  alidade  i8o°,  re- 
level, read  the  vertical  angle  again  with  telescope  in  a reversed 
position,  and  take  the  mean.  This  can  only  be  done  in  case 
the  vertical  limb  is  a complete  circle.  In  many  instruments  it  is 
but  a half-circle  or  less,  in  which  case  this  elimination  cannot 
be  made.  The  accuracy  of  the  adjustments  alone  can  then 
be  relied  on,  and  these  must  be  frequently  tested.  If  the  plate- 
bubble  parallel  to  the  vertical  circle,  the  telescope-bubble,  and 
the  vernier  of  the  vertical  circle  have  all  been  once  accurately 
adjusted,  then  when  these  bubbles  are  brought  to  a zero-read- 
ing the  vertical  circle  should  also  read  zero.  This  test  can 
always  be  readily  applied,  and,  though  not  an  absolute  check, 
it  is  a very  good  one,  inasmuch  as  two  of  these  three  adjust- 
ments would  have  to  be  out  by  the  same  amount  and  in  the 
same  direction  to  still  agree  with  the  third. 

100.  To  run  out  a Straight  Line.— The  transit-instru- 
ment is  especially  adapted  to  the  prolongation  of  straight  lines, 
as  long  tangents  on  railroads,  and  yet  it  requires  the  most  care- 
ful work  and  much  repetition  to  run  a line  that  approximates 
very  closely  to  a straight  line. 

Having  determined  the  direction  which  the  line  is  to  take 
from  the  initial  point,  set  accurately  over  this  point,  turn  the 
telescope  in  the  given  direction,  and  set  a second  point  at  a 
convenient  distance.  These  two  points  now  determine  the 
line,  and  it  remains  to  prolong  it  indefinitely  over  such  uneven 
ground  as  may  lie  in  its  course.  The  line,  when  established, 
is  to  be  the  trace  of  a vertical  plane  through  the  first  two 
points  on  the  surface  of  the  ground.  If  the  line  of  collimation 
always  revolved  in  a vertical  plane,  and  no  errors  were  made 
in  handling  the  instrument  and  in  setting  the  points,  the  prob- 
lem would  be  easily  solved,  but  we  may  safely  say  that  the 
surface  generated  by  the  line  of  collimation  never  is  a vertical 
plane.  (The  adjustments  being  never  absolutely  correct.) 


96 


SC/A'  VE  YING. 


This  surface  is  a cone  whose  axis  is  not  strictly  horizontal,  for 
both  the  horizontal  and  vertical  axes  are  somewhat  inclined 
from  their  true  positions.  It  remains  then  so  to  make  the 
observations  that  all  these  errors  of  adjustment  will  be  elimi- 
nated. The  following  programme  accomplishes  this  : 

(1)  Set  accurately  over  the  forward  point,  putting  one  pair 
of  levelling-screws  in  the  line. 

(2)  Clamp  the  horizontal  limb  in  any  position. 

(3)  Level  carefully,  and  turn  upon  the  rear  point. 

(4)  Relevel  for  the  bubble  that  lies  across  the  line. 

(5)  Make  the  bisection  on  the  rear  point,  revolve  the  tele- 
scope, and  set  a point  in  advance.  This  may  be  a tack  in  a 
stake  set  with  great  care  by  making  the  bisection  on  a pencil 
held  vertically  on  the  stake. 

(6)  Unclamp  the  alidade  and  revolve  it  about  the  vertical 
axis  till  the  telescope  comes  on  the  rear  point. 

(7)  Relevel  for  the  cross  bubble  again. 

(8)  Make  the  bisection  on  the  rear  point,  revolve  the  tele- 
scope again,  and  set  a second  point  in  advance  beside  the  first 
one.  The  mean  of  these  two  positions  should  lie  in  the  verti- 
cal plane  through  the  two  established  points,  whatever  may  be 
its  elevation,  and  regardless  of  small  errors  in  the  instrumental 
adjustments.  For  the  reversals  of  the  telescope  and  alidade 
eliminated  the  errors  of  collimation  and  horizontal  axis,  while 
the  relevelling  eliminated  the  error  due  to  the  error  of  adjust- 
ment of  the  plate-bubble.  If  this  bubble  were  out  of  adjust- 
ment the  vertical  axis  inclined  as  much  to  one  side  for  the  first 
setting  as  it  did  to  the  other  side  for  the  second  setting. 

This  operation  may  be  repeated  for  a check,  or  to  further 
eliminate  errors  of  observation.  The  instrumental  errors  are 
wholly  eliminated  by  one  set  of  observations,  as  above  given. 
It  will  be  noted  that  this  method  is  independent  of  the  gradua- 
tion of  the  limb.  The  only  assumptions  are  that  the  instru- 
ment and  its  adjustments  are  rigid  during  the  reversal  of  the 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  97 


telescope,  and  that  the  pivots  of  the  horizontal  axis  are  true 
cylinders. 

loi.  Traversing. — A traverse,  in  surveying,  is  a series  of 
consecutive  courses  whose  lengths  and  bearings,  or  azimuths, 
have  been  determined.  When  a compass  is  used  the  bearing 
of  each  course  is  determined  by  the  needle  independently  of 
that  of  the  preceding  course.  When  a transit  is  used  and  the 
needle  not  read,  the  graduated  circle  of  the  instrument  is 
always  oriented,  or  brought  into  the  meridian,  by  taking  a 
back-sight  to  the  preceding  station.  If  the  azimuth*  of  the 
first  course  is  known  with  reference  to  the  meridian,  the 
azimuth  of  all  subsequent  courses  may  be  at  once  determined 
by  properly  orienting  the  limb  of  the  instrument  at  the  suc- 
cessive stations.  Thus,  if  the  south  point  has  a zero  azimuth 
the  limb  of  the  instrument  should  be  oriented  at  each  station, 
so  that  when  the  telescope  points  south  vernier  A shall  read 
zero. 

The  forward  azimuth  of  a line  is  its  angular  deviation  from 
the  south  point  when  measured  at  the  rear  station  forward 
along  the  line. 

The  back  azimuth  of  a line  is  its  angular  deviation  from  the 
south  point  at  the  forward  station  when  measured  from  that 
station  back  along  the  line. 

The  forward  and  back  azimuth  differ  by  i8o°  plus  or  minus 
the  convergence  of  the  meridians  at  the  two  extremities  of  the 
line.  If  this  line  is  north  and  south  it  lies  in  the  meridian, 
and  hence  its  forward  and  back  azimuth  differ  by  i8o°.  When 
the  course  has  an  easterly  or  westerly  component,  or,  in  other 
words,  when  its  extremities  have  different  longitudes,  the 
divergence  of  the  line  from  the  meridian  at  one  end  differs 
from  its  divergence  from  the  meridian  at  the  other  by  as  much 

* In  this  treatise  azimuth  is  always  reckoned  from  the  south  point  in  the 
direction  S.W.N.E.  to  360°.  The  bearing  of  the  line  is  thus  given  by  its 
numerical  value  alone,  without  the  aid  of  letters. 

7 


98 


SURVEYING. 


as  these  meridians  differ  from  parallelism.  This  is  inappre- 
ciable on  short  lines,  and  hence  in  traversing  the  forward  and 
back  azimuth  will  be  considered  as  dihering  by  i8o°. 

The  field-work  proceeds  as  follows,  so  far  as  the  transit  is 
concerned.  Let  it  be  assumed  that  from  the  initial  point  A 
of  the  survey  the  true  azimuth  to  some  other  point  Z is  given. 
Let  the  stations  be  A,  B,  C,  etc. 

Set  vernier  A to  read  the  known  azimuth  AZ.  With  the 
alidade  and  limb  clamped  together,  turn  the  telescope  on  Z 
and  clamp  the  limb,  setting  carefully  by  means  of  the  lower 
tangent-screw.  If  the  alidade  be  now  loosened  and  vernier  A 
made  to  read  zero,  the  telescope  would  point  south.  Turn 
the  telescope  on  B by  moving  the  alidade  alone,  and  the  read- 
ing of  vernier  A gives  the  forward  azimuth  of  the  line  AB. 
Move  the  instrument  to  B and  set  vernier  A to  read  the  back 
azimuth  of  AB,  which  is  found  by  adding  i8o"  to  or  subtract- 
ing it  from  the  forward  azimuth,  according  as  this  was  less  or 
more  than  i8o°.  With  alidade  and  limb  clamped  at  this  read- 
ing, turn  upon  A,  clamp  the  limb  and  unclamp  the  alidade,  and 
the  instrument  is  again  properly  oriented  for  reading  directly 
the  true  azimuth  of  any  line  from  this  station,  as  the  line  BC, 
for  instance.  In  this  manner  a traverse  lyiay  be  run  with  the 
transit,  the  field-notes  showing  the  true  azimuth  of  each  course 
without  reduction.  The  lengths  of  the  courses  may  be  found 
in  any  manner  desired. 

If  preferred,  the  telescope  may  be  revolved  on  its  horizon- 
tal axis  and  vernier  A left  with  its  forward  reading,  for  orient- 
ing. Then  revolve  the  telescope  back  to  its  normal  position 
and  proceed  with  the  work.* 


* For  a method  of  computing  the  coordinates  of  the  courses,  and  the  use  of 
the  traverse  table,  see  chapter  on  Land  Surveying. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  QQ 


THE  SOLAR  ATTACHMENT. 

102.  The  Solar  Attachment  is  a device  to  be  fastened  to 
the  telescope  axis  of  a transit-instrument,  thus  making  a com- 
bination that  will  do  the  work  of  a solar  compass.  One  form 
of  this  device  is  shown  in  Fig.  The  various  spherical 

functions  concerned  in  the  problem  are  also  represented  in  this 
figure  by  their  several  great  circles.  The  polar  axis,  declination- 
arc,  and  collimation-arm  are  the  same  here  as  in  the  solar  com- 
pass. The  latitude-arc  is  here  replaced  by  the  vertical  circle 
of  the  transit,  and  the  telescope  gives  the  line  of  sight.  The 
adjustments  and  working  of  this  attachment  are  so  nearly  iden- 
tical with  those  of  the  solar  compass  that  they  will  not  be 
repeated  here.  If  the  student  has  mastered  the  principles 
involved  in  the  use  of  the  solar  compass  he  will  have  no  diffi- 
culty in  using  the  attachment. 

Various  forms  of  solar  attachments  have  been  invented,  the 
most  recent  and  perhaps  the  most  efficient  of  which  is  that 
shown  in  Fig.  20,  invented  by  G.  N.  Saegmuller  in  1881.  It 
is  manufactured  by  Fauth  & Co.,  Washington,  D.  C.,  and  by 
Keuffel  & Esser,  New  York.  It  consists  simply  of  an  auxiliary 
telescope  with  bubble  attached,  having  two  motions  at  right 
angles  to  each  other.  These  motions  are  horizontal  and  verti- 
cal when  the  main  telescope,  to  which  the  attachment  is  rigidly 
fastened,  is  horizontal.  If  the  main  telescope  be  put  in  the 
meridian  and  elevated  into  the  plane  of  the  celestial  equator, 
however,  then  the  vertical  axis  of  the  attachment  also  lies  in 
the  meridian  but  points  to  the  pole.  It  therefore  becomes  a 
polar  axis  about  which  the  auxiliary  telescope  may  revolve.  If 
this  telescopic  line  of  sight  be  at  right  angles  to  the  polar  axis, 
it  will  generate  an  equatorial  plane.  If  the  line  of  sight  be  in- 
clined to  this  plane  by  an  amount  equal  to  and  in  the  direction 
of  the  sun’s  declination,  then  when  revolved  on  its  polar  axis  it 


*From  Gurley’s  Catalogue. 


100 


SUR  VE  YING. 


Fig.  19. 


Fig.  20. 


102 


S UK  VE  Y INC. 


would  follow  the  sun’s  path  in  the  heavens  for  the  given  day, 
provided  the  sun  did  not  change  its  declination  during  the  day. 
It  only  remains,  therefore,  to  show  how  the  latitude  and  decli- 
nation angles  may  be  set  off  in  order  that  the  competency  of 
this  instrument  to  do  the  work  of  the  solar  compass  may  be- 
come apparent. 

To  set  off  the  declination-angle,  turn  the  main  telescope 
down  or  up  according  as  the  declination  is  north  or  south,  and 
set  the  declination-angle  on  the  vertical  circle.  Bring  the 
small  telescope  into  the  plane  of  the  large  one  and  revolve  it 
about  its  horizontal  axis  until  its  bubble  comes  to  the  centre 
of  its  tube.  The  angle  formed  by  the  two  telescopic  lines  of 
sight  is  the  declination-angle.  Revolve  the  main  telescope 
until  it  has  an  altitude  equal  to  the  co-latitude  of  the  place, 
and  clamp  it  in  this  position.  With  the  vertical  motions  of 
both  telescopes  clamped,  and  their  lateral  motions  free,  if  the 
line  of  sight  of  the  small  telescope  can  be  brought  upon  the 
sun  the  main  telescope  must  lie  in  the  meridian.  The  vertical 
circle  of  the  transit  is  thus  seen  to  do  the  work  of  both  the 
latitude  and  declination  arcs  of  the  solar  compass. 

103.  Adjustments  of  the  Saegmuller  Attachment. — 
First,  All  the  adjustments  of  the  transit  must  be  as  perfect 
as  possible,  but  especially  the  plate  and  telescope  bubbles,  the 
vernier  of  the  vertical  circle,  and  the  transverse  axis  of  the 
telescope. 

Second,  To  make  the  Polar  Axis  perpendicular  to  the  Plane 
of  the  Line  of  Collimation  and  Horizontal  Axis  of  the  Main 
Telescope, — Carefully  level  the  instrument  and  bring  the  teles- 
cope-bubble to  the  middle  of  its  tube.  The  line  of  sight  and 
horizontal  axis  of  this  telescope  should  now  be  horizontal,  so 
that  the  polar  axis  is  to  be  made  vertical.  To  test  this,  revolve 
the  auxiliary  telescope  about  the  polar  axis,  and  see  if  the 
bubble  on  the  small  telescope  maintains  a constant  position. 
If  not,  correct  half  the  movement  by  means  of  the  adjusting, 
screws  at  the  base  of  the  small  disk,  and  the  other  half  by  re- 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  IO3 


volving  the  auxiliary  telescope.  These  adjusting-screws  are 
exactly  analogous  to  the  levelling-screws  of  the  main  instru- 
ment. 

Third,  To  make  the  Lme  of  Sight  of  the  Sinall  Telescope 
parallel  to  the  Axis  of  the  Attached  Bubble, — Make  the  large 
telescope  horizontal  by  bringing  its  attached  bubble  to  the 
middle  of  its  tube.  Bring  the  small  telescope  in  the  same  plane 
and  make  it  also  horizontal  by  means  of  its  bubble,  clamping 
its  vertical  motion.  Measure  the  vertical  distance  between  the 
axis  of  the  two  telescopes,  and  lay  off  this  distance  on  a piece  of 
paper  by  two  plain  horizontal  lines.  Set  this  paper  up  at  a con- 
venient distance  from  the  instrument,  and  on  about  the  same 
level.  Bring  the  line  of  sight  of  the  large  telescope  on  the  lower 
mark,  and  see  if  that  of  the  small  telescope  falls  on  the  upper 
mark.  If  not,  adjust  its  reticule  until  its  line  of  collimation 
does  come  on  the  upper  mark.  Revolve  back  to  the  horizontal 
to  see  if  both  bubbles  again  come  to  the  middle  simultaneously. 

When  this  adjustment  is  completed,  there  should  be  five 
lines  in  the  instrument  parallel  to  each  other  when  instrument 
and  telescopes  are  level, — viz.,  the  axes  of  the  two  telescope- 
bubbles  and  of  the  plate-bubble  on  the  standards,  and  the  two 
lines  of  collimation, — and,  in  addition,  the  vernier  on  the  vertical 
circle,  should  read  zero. 

The  seven  adjustments  (five  of  the  transit  and  two  of  the 
attachment)  must  all  be  carefully  made  and  frequently  tested 
if  the  best  results  are  desired.  When  this  is  done,  this  attach- 
ment will  give  the  meridian  to  the  nearest  minute  of  arc,  if  ob- 
servations be  taken  when  the  sun  is  more  than  one  hour  from 
the  horizon  and  two  hours  from  the  meridian.  The  advantages 
of  the  Saegmuller  attachment  consist  mainly  in  having  a teles- 
copic line  of  sight,  and  in  the  use  of  the  vertical  limb  of  the 
transit  for  setting  off  the  declination  and  co-latitude.  The 
effect  of  small  errors  in  the  latitude  and  declination  angles,  such 
as  may  be  due  to  errors  in  the  adjustments,  is  shown  by  the 
table,  art.  54,  p.  51. 


104 


SURVEYING. 


THE  GRADIENTER  ATTACHMENT. 

104.  The  Gradienter  is  a tangent-scrcw  with  a micrometer- 
head  attached  to  the  horizontal  axis  of  the  telescope  for  the 
purpose  of  turning  off  vertical  angles  that  are  expressed  in 
terms  of  its  tangent  as  so  many  feet  to  the  hundred.  Such  a 
device  is  shown  in  Fig.  17.  In  railroad  work,  the  grade  or  slope 
is  expressed  in  this  manner,  as  26.4  feet  per  mile,  or  as  0.5  foot 
per  ICO  feet.  The  micrometer-head  is  graduated  so  that  one 
revolution  raises  or  lowers  the  telescope  by  i foot  or  0.5  foot 
in  100  feet.  It  is  divided  into  100  or  50  parts,  so  that  each 
division  on  the  head  is  equivalent  to  o.oi  foot  in  100  feet.  This 
attachment  is  found  very  convenient  in  railroad  work.  It  is 
also  of  general  utility  in  obtaining  approximate  distances.  On 
level  ground  the  distance  is  read  directly,  but  on  sloping 
ground  the  rod  is  still  held  vertical,  and  the  distance  read  is  too 
great.  The  true  horizontal  distance  may  be  found  by  multiply- 
ing the  distance  read  by  the  factors  for  horizontal  distance 
given  in  table  V.*  Thus,  if  one  revolution  of  the  screw  raises 
the  line  of  sight  i foot  at  a distance  of  100  feet,  and  if  at  a cer- 
tain unknown  distance  one  revolution  of  screw  caused  the  line 
to  pass  over  5.5  feet  on  the  rod,  then  the  distance  was  550  feet 
if  the  ground  was  horizontal.  If  the  rod-readings  had  a mean 
vertical  angle  of  15°,  the  horizontal  distance  was  550  X 93.3  = 
513  feet. 


CARE  OF  THE  TRANSIT. 

105.  The  Transit  should  be  protected  from  rain  and  dust 
as  much  as  possible.  A silk  gossamer  water-proof  bag  should 
be  carried  by  the  observer  to  be  used  for  this  purpose.  If  water 
gets  inside  the  telescope,  remove  the  eye-piece  and  let  it  dry 
out.  If  moisture  collects  between  the  two  parts  of  the  objec- 


* This  table  is  for  reduction  of  stadia  measurements,  and  is  explained  in 
the  chapter  on  Topographical  Surveying,  p.  233. 


ADJUSTMENT,  USE,  AND  CARE  OF  INS7EUMENTS.  IO5 


tive  remove  it,  and  dry  it  with  a gentle  heat  over  a stove  or 
lamp,  but  do  not  separate  the  glasses.  If  dust  settles  on  the  wires 
it  may  be  blown  off  by  removing  both  objective  and  eye-piece 
and  blowing  gently  through  the  tube.  Dust  should  be  removed 
from  the  glasses  by  a camel’s-hair  brush,  which  should  always 
be  carried  for  the  purpose.  A clean  handkerchief  maybe  used 
with  a gentle  pressure  to  prevent  scratching  in  case  the  dust  is 
gritty.  Use  alcohol  for  cleansing  greasy  or  badly  soiled  glasses. 
No  part  exposed  to  dust  should  be  oiled,  as  this  serves  to  retain 
all  the  dust  that  may  fall  on  it.  The  centres  should  be  cleaned 
occasionally  with  chamois  skin,  and  oiled  by  a very  little  pure 
watch-oil.  In  the  absence  of  watch-oil  plumbago  will  be  found 
to  serve.  A soft  lead-pencil  may  be  scraped  and  a little  rubbed 
on  the  spindles  with  the  finger.  The  tripod  legs  should  have 
no  lost  motion  either  at  the  head  or  in  their  iron  shoes.  If  the 
legs  are  split,  as  in  Fig.  17,  and  fastened  by  thumb-nuts,  these 
should  be  loosened  when  the  instrument  is  carried  and  tight- 
ened again  after  setting.  They  may  thus  be  made  very  tight  and 
rigid  while  the  instrument  is  in  use  without  danger  of  break- 
ing the  bolts  in  closing  the  legs,  which  is  very  liable  to  result 
if  the  screws  are  not  loosened.  For  a method  of  putting  in  new 
cross-wires  see  chapter  on  Topographical  Surveying,  p.  236. 

EXERCISES  WITH  THE  TRANSIT. 

106.  Establish  three  stations  forming  a triangle.  Measure  the  three  hori- 
zontal angles  and  see  if  their  sum  is  180°. 

107.  Prolong  a line  in  azimuth  and  distance  by  carrying  both  around  an 
imaginary  obstruction,  and  then  check  the  azimuth  by  a back-sight  and  the  dis- 
tance by  measurement.  Thus,  let  A and  B be  two  points  establishing  a line. 
The  problem  is  to  establish  two  other  points,  C and  D,  in  the  continuation  of 
the  line  AB,  with  an  imaginary  obstruction  to  both  sight  and  measurement 
between  B and  C.  The  distance  BC  is  also  to  be  obtained. 

The  equilateral  triangle  will  be  found  most  efficient. 

108.  Find  both  the  distance  to  and  the  height  of  an  inaccessible  steeple, 
chimney,  smokestack,  or  tree. 

Measure  a base-line  such  that  its  two  extremities  make  with  the  given  object 


io6 


SUR  VE  YING. 


% 


approximately  an  isosceles  triangle  (it  is  desirable  that  no  angle  of  the  triangle 
should  be  less  than  30®  nor  more  than  120°).  The  top  of  the  object  only  need 
be  visible  from  the  two  ends  of  the  base.  Measure  both  the  horizontal  and 
vertical  angles  at  the  extremities  of  the  base-line  subtended  by  the  other  two 
points  of  the  triangle.  Let  A and  B be  the  extremities  of  the  base  and  P the 
point  whose  distance  and  elevation  are  required.  We  then  have  for  horizontal 
angles 

Sin  P ; sin  A ::  AB  : BP\ 
also  sin  P : sin  B ::  AB  : AP. 

In  reading  the  vertical  angles  to  the  base-stations  the  reading  should  be 
taken  on  a point  as  high  above  the  ground  (or  peg)  as  the  telescope  is  above  the 
peg  over  which  it  is  set.  The  difference  in  the  elevations  of  the  two  pegs  is 
then  obtained.  The  vertical  angle  to  the  point  P is  taken  to  the  summit,  and 
height  of  instrument  added  in  each  case  to  find  its  elevation  above  peg.  If  A 
be  the  lower  of  the  two  base-stations  and  if  I a and  lu  be  the  heights  of  instru- 
ment (line  of  sight)  above  the  peg  in  the  two  cases,  and  if  Va,  Vb,  Fpand 
Vp'  be  the  vertical  angles  read  to  the  corresponding  points,  we  may  write: 

Elevation  of  B above  A = AB  tan  Vb\ 

“ “ />  *«  ^ 

Also,  from  the  vertical  angles  taken  at  B,  we  have : 

Elevation  of  A below  B = AB  tan  Va\ 

“ “ above  tan  Vp'. 

We  now  have  a check  on  both  the  relative  elevations  and  on  the  distances 
AP  and  BP.  Assuming  the  elevation  of  A to  be  zero,  we  have: 

Elevation  of  P above  A — AP  tan  Vp  = AB  tan  Vb  + BP  tan  Vp'. 

This  equality  will  not  result  unless  the  observations  were  well  taken,  the 
computations  accurately  made,  and  the  instrument  carefully  adjusted.  The  ad- 
justments mainly  involved  here  are  the  plate-bubbles  and  the  vernier  on  the 
vertical  circle.  If  the  points  are  a considerable  distance  apart,  as  over  a half- 
mile,  the  elevations  obtained  by  reading  the  vertical  angles  are  appreciably  too 
great,  on  account  of  the  earth’s  curvature.  This  may  be  taken  as  eight  inches 
for  one  mile  and  proportional  to  the  square  of  the  distance.  Or,  we  may  write: 

Elevation  correction  on  long  sights,  in  inches,*  = — 8 (distance  in  miles'!®. 

If  the  distances  are  all  less  than  about  half  a mile,  no  attention  need  be  paid 
to  this  correction  in  this  problem. 


For  a full  discussion  of  this  subject  see  chap.  XIV. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  I07 


109.  Find  the  height  of  a tree  or  house  above  the  ground,  on  a distant  hill, 
without  going  to  the  immediate  locality. 

no.  Find  the  horizontal  length  and  bearing  of  a line  joining  two  visible  but 
inaccessible  objects.  Use  the  magnetic  bearing  if  the  true  bearing  of  the  base- 
line is  not  known.  ^ 

11^  Find  the  horizontal  length  and  bearing  of  a line  joining  two  inaccessi- 
ble points  both  of  which  cannot  be  seen  from  any  one  position. 

Let  A and  B be  the  inaccessible  points.  Measure  a base  CD  such  that  A is 
seen  from  C,  and  B from  D.  Auxiliary  bases  and  triangles  may  be  used  to 
find  the  lengths  of  ^ C and  BD.  Knowing  A C and  CD  and  the  included  angle, 
compute  AD  in  bearing  and  distance.  The  angle  ADB  may  now  be  found, 
which,  with  the  adjacent  sides  AD  and  BD  known,  enables  the  side  AB  to  be 
found  in  bearing  and  distance. 

1 12.  With  the  transit  badly  out  of  level,  or  with  horizontal  axis  of  the  tele- 
scope thrown  considerably  out  of  the  horizontal,  measure  the  horizontal  angle 
between  two  objects  having  very  different  angular  elevations.  Do  this  with 
both  telescope  normal  and  telescope  reversed,  and  note  the  difference  in  the 
values  of  the  angle  obtained  in  the  two  cases. 

1 13.  Select  a series  of  points  on  uneven  ground,  enclosing  an  area,  and 
occupy  them  successively  with  the  transit,  obtaining  the  traverse  angles.  That 
is,  knowing  or  assuming  the  azimuth  of  the  first  line,  obtain  the  azimuths  of  the 
other  connecting  lines,  or  courses,  with  reference  to  this  one,  returning  to  the 
first  point  and  obtaining  the  azimuth  of  the  first  course  as  carried  around  by  the 
traversed  line.  This  should  agree  with  the  original  azimuth  of  this  course. 
The  distances  need  not  be  measured  for  this  check. 

1 14.  Lay  out  a straight  line  on  uneven  ground  by  the  method  given  in  Art. 
100,  occupying  from  six  to  ten  stations.  Return  over  the  same  line  and  estab- 
lish a second  series  of  points,  paying  no  attention  to  the  first  series,  and  then 
note  the  discrepancies  on  the  several  stakes.  In  returning,  the  two  final  points 
of  the  first  line  become  the  initial  points  of  the  second,  this  return  line  being  a 
prolongation  of  the  line  joining  these  two  points.  If  these  deviate  ever  so 
little,  therefore,  from  the  true  line,  the  discrepancy  will  increase  towards  the 
initial  point. 

Similar  exercises  to  those  given  for  the  solar  compass  may  be  assigned  for 
the  solar  attachment. 


io8 


SUR  VE  YING. 


on  board  ship.  It  is  exclusively  used  in  observations  at  sea, 
and  is  always  used  in  surveying  where  angles  are  to  be  meas- 
ured from  a boat,  as  in  locating  soundings,  buoys,  etc.,  as  well 
as  in  reconnoissance  work,  explorations,  and  preliminary  sur- 
veys. It  has  been  in  use  since  about  1730. 

The  accompanying  cut  shows  a common  form  of  this  in- 
strument as  manufactured  by  Fauth  & Co.,  Washington.  The 
limb  has  a 7|-inch  radius,  and  reads  to  10  seconds  of  arc. 


THE  SEXTANT. 

I15.  The  Sextant  is  the  most  convenient  and  accurate 
hand-instrument  yet  devised  for  measuring  angles,  whether 
horizontal,  vertical,  or  inclined.  It  is  called  a sextant  because 
its  limb  includes  but  a 60°  arc  of  the  circle.  It  will  measure 
angles,  however,  to  120°.  It  is  held  in  the  hand,  measures  an 
angle  by  a single  observation,  and  will  give  very  accurate  re- 
sults even  when  the  observer  has  a very  unstable  support,  as 


Fig.  21. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  IO9 


There  is  a mirror  7^/(Fig.  22),  called  the  Index  Glass,  rigidly- 
attached  to  the  movable  arm  MA,  which  carries  a vernier 
reading  on  the  graduated  limb  CD.  There  is  another  mirror, 
I,  called  the  Horizon  Glass,  rigidly  attached  to  the  frame  of 
the  instrument,  and  a telescope  pointing  into  this  mirror,  also 
rigidly  attached.  This  mirror  is  silvered  on  its  lower  half,  but 
clear  on  its  upper  half.  A ray  of  light  coming  from  H passes 


through  the  clear  portion  of  the  mirror  / on  through  the  tele- 
scope to  the  eye  at  E.  Also,  a ray  from  an  object  at  O strikes 
the  m\xxox  M,  is  reflected  to  m,  and  then  through  the  telescope 
to  E.  Through  one  half  of  the  objective  come  the  rays  from 
H,  and  through  the  other  half  the  rays  from  O,  each  of  which 
sets  of  rays  forms  a perfect  image.  By  moving  the  arm  MA 
it  is  evident  these  images  will  appear  to  move  over  each  other, 


no 


SUR  VE  YING. 


and  for  one  position  only  will  they  appear  to  coincide.  The 
bringing  of  the  two  images  into  exact  coincidence  is  what  the 
observation  consists  in,  and  however  unsteady  the  motion  of 
the  observer  may  be,  he  can  occasionally  see  both  images  at 
once,  and  so  by  a series  of  approximations  he  may  finally  put 
the  arm  in  its  true  position  for  exact  superposed  images. 
The  angle  subtended  by  the  two  objects  is  then  read  off  on 
the  limb. 

Ii6.  The  Theory  of  the  Sextant  rests  on  the  optical 
principle  that  “ if  a ray  of  light  suffers  two  successive  reflec- 
tions in  the  same  plane  by  two  plane  mirrors,  the  angle  be- 
tween the  first  and  last  directions  of  the  ray  is  twice  the  angle 
of  the  mirrors.” 

To  prove  this,  let  (9J/and  mEhe  the  first  and  last  posi- 
tions of  the  ray,  the  latter  making  with  the  former  produced 
the  angle  E.  The  angle  of  the  mirrors  is  the  angle  A.  The 
angles  of  incidence  and  reflection  at  the  two  mirrors  are  the 
angles  i and  PM,  and  p7n  being  the  normals. 

We  may  now  write  : 

Angle  E = OMm  — MmE, 

— lit  ^ j I 

angle  A = ImM  — mMA 

= (9o°-O-(9O°-0 

= i—  t\ 

Therefore  E = 2A.  Q.  E.  D. 

When  the  mirrors  are  brought  into  parallel  planes,  the 
angle  A becomes  zero,  whence  E also  is  zero,  or  the  rays  OM 
and  Hin  are  parallel.  This  gives  the  position  of  the  arm  for 
the  zero-reading  of  the  vernier.  The  limb  is  graduated  from 
this  point  towards  the  left  in  such  a way  that  a 60°  arc  of  the 
circle  will  read  to  120°.  That  is,  a movement  of  1°  on  the  arc 
really  measures  an  angle  of  2°  in  the  incident  rays,  so  it  must 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  Ill 


be  graduated  as  two  degrees  instead  of  one.  The  very  large 
radius  enables  this  to  be  done  without  difficulty. 

ADJUSTMENTS  OF  THE  SEXTANT. 

117.  To  make  the  Index  Glass  perpendicular  to  the 
Plane  of  the  Sextant. — Bring  the  vernier  to  read  about  30° 
and  examine  the  arc  and  its  image  in  the  index  glass  to  see  if 
they  form  a continuous  curve.  If  the  glass  is  not  perpendi- 
cular to  the  plane  of  the  arc,  the  image  will  appear  above  or 
below  the  arc,  according  as  the  mirror  leans  forward  or  back- 
ward. It  is  adjusted  by  slips  of  thin  paper  under  the  project- 
ing points  and  corners  of  the  frame. 

118.  To  make  the  Horizon  Glass  Parallel  to  the  Index 
Glass  for  a Zero-reading  of  the  Vernier. — Set  the  vernier 
to  read  zero  and  see  if  the  direct  and  reflected  images  of  a 
well-defined  distant  object,  as  a star,  come  into  exact  coinci- 
dence. If  not,  adjust  the  horizon  glass  until  they  do.  If  this 
adjustment  cannot  be  made,  bring  the  objects  into  coincidence, 
or  even  with  each  other  so  far  as  the  motion  of  the  arm  is  con- 
cerned, and  read  the  vernier.  This  is  the  index  error  of  the 
instrument  and  is  to  be  applied  to  all  angles  read.  The  better 
class  of  instruments  all  allow  the  horizon  glass  to  be  adjusted. 
This  adjustment  is  generally  given  as  two,  but  it  is  best  con- 
sidered as  one.  If  made  parallel  to  the  index  glass  after  that 
has  been  adjusted,  it  must  be  perpendicular  to  the  plane  of 
the  instrument. 

119.  To  make  the  Line  of  Sight  of  the  Telescope 
parallel  to  the  Plane  of  the  Sextant. — The  reticule  in  the 
sextant  carries  four  wires  forming  a square  in  the  centre  of 
the  field.  The  centre  of  this  square  is  in  the  line  of  collima- 
tion  of  the  instrument. 

Rest  the  sextant  on  a plane  surface,  pointing  the  telescope 
upon  a well-defined  point  some  twenty  feet  distant.  Place  two 
objects  of  equal  height  upon  the  extremities  of  the  limb  that 


II2 


SUI^  VE  YING. 


will  serve  to  establish  a line  of  sight  parallel  to  the  limb.  Two 
lead-pencils  of  same  diameter  will  serve,  but  they  had  best  be 
of  such  height  as  to  make  this  line  of  sight  even  with  that  of 
the  telescope.  If  both  lines  of  sight  come  upon  the  same 
point  to  within  a half-inch  or  so  at  a distance  of  20  feet, 
the  resulting  maximum  error  in  the  measurement  of  an  angle 
will  be  only  about  i". 

THE  USE  OF  THE  SEXTANT. 

120.  To  measure  an  Angle  with  the  sextant,  bring  its 
plane  into  the  plane  of  the  two  objects.  Turn  the  direct  line 
of  sight  upon  the  fainter  object,  which  may  require  the  instru- 
ment to  be  held  face  downwards,  and  bring  the  two  images 
into  coincidence.  The  reading  of  the  limb  is  the  angle  re- 
quired. It  must  be  remembered  that  the  angles  measured  by 
the  sextant  are  the  true  angles  subtended  by  the  two  objects  at 
the  point  of  observation,  and  not  the  vertical  or  horizontal 
projection  of  these  angles,  as  is  the  case  with  the  transit.  The 
true  vertex  of  the  measured  angle  is  at  E,  Fig.  21.  It  is  evident 
the  position  of  E is  dependent  on  the  size  of  the  angle,  being 
at  a great  distance  back  of  the  instrument  for  a very  small 
angle.  The  instrument  should  therefore  not  be  used  for  meas- 
uring very  small  angles  except  as  between  objects  a very  great 
distance  off.  The  sextant  is  seldom  or  never  used  for  measur- 
ing angles  where  the  position  of  the  instrument  (or  the  vertex 
of  the  angle)  needs  to  be  known  with  great  accuracy. 

EXERCISES  FOR  THE  SEXTANT. 

121.  Measure  the  altitude  of  the  sun  or  a star  at  its  culmination  by  bringing 
the  direct  image,  reflected  from  the  surface  of  mercury  held  in  a flat  dish  on 
the  ground,  into  coincidence  with  the  image  reflected  from  the  index  glass. 
Half  the  observed  angle  is  the  altitude  of  the  body.  The  altitude  of  a terres- 
trial object  may  be  obtained  in  the  same  manner,  in  which  case  the  vessel  of 
mercury  should  rest  on  an  elevated  stand  ; the  sextant  could  then  be  brought 
near  to  it  and  the  angular  divergence  of  the  two  incident  rays  to  the  mercury 
surface  and  index  glass  reduced  to  an  inappreciable  quantity. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  II3 


If  the  observation  of  a heavenly  body  be  made  on  the  meridian  and  the 
declination  of  the  body  at  the  time  of  observation  be  known,  the  latitude  of  the 
place  is  readily  found. 

I2ia.  Measure  the  angle  subtended  by  two  moving  bodies,  as  of  two  men 
walking  the  street  in  the  same  direction,  or  of  two  boats  on  the  water.  (This  is 
to  illustrate  the  capacity  of  the  sextant,  for  none  but  a reflecting  instrument 
bringing  two  converging  lines  of  sight  into  coincidence  is  competent  to  do 
this.) 

The  exercises  given  in  Arts.  106,  108,  109,  and  no  for  the  transit  may  also 
serve  for  the  sextant.  Further  applications  of  the  sextant  in  locating  soundings 
are  given  in  chap.  X. 

122.  The  Double-reflecting  Gpniograph  is  a kind  of  dou- 
ble sextant  and  three-arm  protractor*  combined.  It  enables  the 
two  variable  angles  of  the  “ three-point  problem”  f to  be 
measured  at  once,  and  then  provides  for  the  immediate  plot- 
ting of  these  angles  upon  the  sheet,  without  reading  off  the 
values  of  the  angles  unless  they  are  to  be  put  on  record.  The 
angles  may  be  read,  however,  and  plotted  afterwards  if  de- 
sired. This  very  ingenious  and  convenient  instrument  is  the 
invention  of  Lieutenant  Constantin  Pott,  of  the  English  Navy. 
The  construction  and  principles  of  the  instrument  are  shown 
in  Figs.  23,  24,  and  25.  To  the  graduated  circle  whose  centre 
is  D,  Fig.  24,  there  are  attached  one  fixed  and  two  movable 
arms,  each  having  one  radial  fiducial  edge.  The  main  frame- 
work of  the  instrument  lies  on  the  prolongation  of  the  fixed 
arm  A.  Immediately  back  of  the  centre  of  the  circle  is  a 
cylindrical  frame  containing  two  fixed  mirrors,  s s,  one  above 
the  other,  and  also  a free  opening,  W,  Fig.  23.  These  corre- 
spond to  the  fixed  mirror  and  clear  glass  on  the  sextant.  Im- 
mediately back  of  these  mirrors  is  the  telescope,  P,  and  on 
each  side  of  this  is  a movable  mirror,  55,  attached  to  the  slide 
bars  //.  These  bars  are  fastened  to  the  mirrors  and  slide 
freely  through  the  studs  Z set  upon  the  movable  arms  B P,. 

* For  a description  of  the  three  arm  protractor,  see  chapter  VI.,  p.  167. 

f See  chapter  X.,  p.  280,  for  a discussion  of  this  problem. 

8 


IT4 


SURVEYING. 


The  distance  of  these  studs  from  the  centre  of  the  graduated 
circle  is  the  same  as  that  of  the  axes  of  the  movable  mirrors 
5 S.  Therefore  a circle  whose  centre  coincides  with  the  centre 
of  the  graduated  circle  may  pass  through  these  four  axes. 


The  theory  "of  the  instrument  is  shown  in  Fig.  25.  The  ray 
of  light  R is  reflected  from  c to  ^ and  thence  down  the  tele- 
scope to  A.  The  object  in  the  prolongation  of  AB  casts  the 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  II5 


ray  Be  directly  down  the  telescope.  The  angle  formed  by 
the  incident  and  final  reflected  ray,  Rfe^  is  twice  the  angle 
subtended  by  the  planes  of  the  mirrors  C g e,  as  was  shown  in 
the  case  of  the  sextant.  When  the  rays  R and  B coincide  the 
mirrors  S S and  s s,  Fig.  24,  are  parallel.  The  slide-bar  then 
has  the  position  Ca.  When  the  arm  has  come  into  the  posi- 
tion dB'f  making  the  angle  0 with  the  fixed  arm  dB^  the  slide- 


bar  has  come  into  the  position  Ca\  making  an  angle  |-0  with 
its  former  position  ca,  since  this  is  an  angle  in  the  circumfer- 
ence. The  mirror  has  also  turned  through  an  angle 
and  since  it  was  parallel  to  the  mirror  ss  in  its  first  position  it 
now  makes  an  angle  y — \ with  it.  The  angle  which  is 
the  angle  subtended  by  the  incident  ray  Rc  and  by  the  direct 
ray  BA,  is  therefore  equal  to  the  angle  0,  which  is  the  angle 
ada'  read  on  the  graduated  circle. 


ii6 


SUR  VE  YING. 


Both  movable  arms  are  provided  with  clamp-screws,  K Ky 
and  tangent  screws,  M M.  The  instrument  is  held,  while 
observing,  by  the  handle  Hy  Fig.  23 ; but  when  used  for  plot- 
ting the  point  of  observation  this  handle  is  unshipped  and  the 
instrument  manipulated  by  the  two  milled  heads  F and  6*. 
The  centre  at  dy  Fig.  24,  is  open,  so  that  when  the  instrument 
is  adjusted  to  the  plotted  positions  of  the  three  known  stations, 
the  point  of  observation  is  marked  by  a pencil  through  the 
open  centre.  It  is  therefore  a double  sextant  for  observing 
and  a three-arm  protractor  for  plotting. 


ADJUSTMENT,  USE,  AND  CADE  OF  INSTRUMENTS,  llj 


CHAPTER  V. 

THE  PLANE  TABLE. 

123.  The  Plane  Table  consists  of  a drawing-board 
properly  mounted  on  which  rests  an  alidade  carrying  a line  of 
sight  rigidly  attached  to  a plain  ruler  with  a fiducial  edge. 
The  line  of  sight  is  usually  determined  by  a telescope,  as  in 
Fig.  26.  This  telescope  has  no  lateral  motion  with  respect  to 
the  ruler,  but  both  may  be  moved  at  pleasure  on  the  table. 
The  telescope  has  a vertical  motion  on  a transverse  axis,  as  in 
the  transit.  It  is  also  provided  with  a level  tube,  either 
detachable  or  permanently  fixed.  The  table  is  levelled  by 
means  of  one  round  or  two  cross  bubbles  on  the  ruler  of  the 
alidade.  The  line  of  sight  of  the  telescope  is  usually  parallel 
to  the  fiducial  edge  of  the  ruler,  though  this  is  not  essential. 
It  is  only  necessary  that  they  should  make  a fixed  horizontal 
angle  with  each  other.  The  table  itself  must  have  a free  hori- 
zontal angular  movement  and  the  ordinary  clamp  and  slow- 
motion  screw.  The  table  corresponds  to  the  graduated  limb 
in  the  transit,  the  alidades  in  the  two  instruments  performing 
similar  duties.  Instead,  however,  of  reading  off  certain  hori- 
zontal angles,  as  is  done  with  the  transit,  and  afterwards 
plotting  them  on  paper,  the  directions  of  the  various  pointings 
are  at  once  drawn  on  the  paper  which  is  mounted  on  the  top 
of  the  table,  no  angles  being  read.  The  true  relative  positions 
of  certain  points  in  the  landscape  are  thus  transferred  directly 
to  the  drawing-paper  to  any  desired  scale.  The  magnetic 
bearing  of  any  line  may  be  determined  by  means  of  the  decli- 
nator,  which  is  a small  box  carrying  a needle  which  can  swing 
some  ten  degrees  either  side  of  the  zero-line.  The  zero-line 


ii8 


SUR  VE  YING. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  I IQ 


being  parallel  to  one  edge  of  the  box,  the  magnetic  meridian 
may  be  at  once  marked  down  on  any  portion  of  the  map,  and 
the  bearing  of  any  intersecting  line  determined  by  means  of  a 
protractor.  The  instrument  has  been  long  and  extensively  used 
for  mapping  purposes,  and  is  still  the  only  instrument  used 
for  the  “ fillingdn”  of  the  topographical  charts  of  the  U.  S. 
Coast  and  Geodetic  Survey.  An  extended  account  of  the 
instrument  and  the  field  methods  in  use  on  that  service  may 
be  found  in  Appendix  13  of  the  Report  of  the  U.  S.  Coast  and 
Geodetic  Survey  for  1880.  The  following  discussion  is  partly 
from  that  source. 

ADJUSTMENTS  OF  THE  ALIDADE. 

124.  To  make  the  Axes  of  the  Plate-bubbles  parallel 
to  the  Plane  of  the  Table. — Level  the  table  with  the  alidade 
in  any  position,  noting  the  readings  of  the  bubbles.  Mark  the 
exact  position  of  the  alidade  on  the  table,  take  it  up  carefully, 
and,  reversing  it  end  for  end,  replace  it  by  the  same  marks.  If 
the  bubbles  now  have  the  same  readings  as  before,  with  refer- 
ence to  the  table  they  are  parallel  to  the  plane  of  the  table. 
If  not,  adjust  the  bubbles  for  one  half  the  movement  and  try 
again. 

125.  To  cause  the  Line  of  Sight  to  revolve  in  a Vertical 
Plane. — This  adjustment  is  the  same  as  in  the  transit.  It  need 
not  be  made  with  such  extreme  accuracy,  however,  and  the 
plumb-line  test  is  sufficient.  With  the  instrument  carefully 
levelled,  cause  the  line  of  sight  to  follow  a plumb-line  through 
as  great  an  arc  as  convenient.  If  the  line  of  sight  deviates 
from  the  plumb-line  raise  or  lower  one  end  of  the  transverse 
axis  of  the  telescope,  until  it  will  follow  it  with  sufficient  exact- 
ness. 

126.  To  cause  the  Telescope-bubble  and  the  Vernier  on 
the  Vertical  Arc  to  read  Zero  when  the  Line  of  Sight  is 
Horizontal. — This  adjustment  is  also  the  same  as  in  the 


120 


SURVEYING. 


transit.  The  methods  given  for  the  transit  may  be  used  with 
the  plane  table,  or  a sea  horizon  may  be  used  as  establishing  a 
horizontal  line,  or  a levelling-instrument  may  be  set  up  beside 
the  plane  table  having  the  telescopes  at  the  same  elevation,  and 
both  lines  of  sight  turned  upon  the  same  point  in  the  horizontal 
plane  as  determined  by  the  level.  The  bubble  and  vernier  are 
then  both  adjusted  to  this  position  of  telescope. 

This  adjustment  is  important  if  elevations  are  to  be  deter- 
mined either  by  vertical  angles  or  by  horizontal  lines  of  sight. 
If  only  geographical  position  is  sought  this  adjustment  may 
be  neglected. 


THE  USE  OF  THE  PLANE  TABLE. 

127.  In  using  the  Plane  Table  at  least  two  points  on  the 
ground,  over  which  the  table  may  be  set,  must  be  plotted  on 
the  paper  to  the  scale  of  the  map  before  the  work  of  locating 
other  points  can  begin.  This  requires  that  the  distance  between 
these  points  shall  be  known,  which  distance  becomes  the  base- 
line for  all  locations  on  that  sheet.  Any  error  in  the  measure- 
ment or  plotting  of  this  line  produces  a like  proportional  error 
in  all  other  lines  on  the  map. 

The  plane  table  is  set  over  one  of  these  plotted  points,  the 
fiducial  edge  of  the  ruler  brought  into  coincidence  with  the  two 
points,  and  the  table  revolved  until  the  line  of  sight  comes  on 
the  distant  point.  The  table  is  now  clamped  and  carefully  set 
by  the  slow-motion  screw  in  this  position,  when  it  is  said  to  be 
oriented,  or  in  position. 

In  Figs.  27  and  28,  let  T,  T,'  T,"  T^"  represent  the  plane- 
table  sheet  and  the  points  a and  p the  original  plotted  points. 
The  corresponding  points  on  the  ground  are  A and  P,  the  latter 
being  covered  by  p in  Fig.  27,  and  the  former  by  in  Fig.  28. 
In  Fig.  27,  the  plotted  point  p is  centred  over  the  point  P,  the 
ruler  made  to  coincide  with  ap,  and  the  telescope  made  to  read 
on  A by  shifting  the  table.  For  plotting  the  directions  of 


I2I 


Fig.  28. 


lO 


122 


SUR  VE  YING. 


other  objects  on  the  ground,  the  alidade  is  made  to  revolve  about 
p just  as  the  transit  revolves  about  its  centre.  A needle  is 
sometimes  stuck  at  this  point,  and  the  ruler  caused  to  press 
against  it  in  all  pointings,  but  this  defaces  the  sheet.  Other 
pointings  are  now  made  to  B,  C,  and  D,  which  may  be  used  as 
stations,  and  also  to  a chimney  (c/l),  a tree  (/.),  a cupola 
{cup.),  a spire  (sp.),  and  a windmill  {w.ml).  Short  lines  are 
drawn  at  the  estimated  distance  from  p,  and  these  marked  with 
letters,  as  in  the  figure,  or  by  numbers,  and  a key  to  the  numbers 
kept  in  the  sketch-  or  note-book. 

The  table  is  now  removed  to  A,  the  other  known  point,  and 
set  with  the  point  a on  the  plot  over  the  point  A on  the 
ground,  when  the  table  is  approximately  oriented.  The  ruler 
is  now  set  as  shown  in  Fig.  28,  coinciding  with  a and  p,  but 
pointing  towards  /.  The  table  is  then  swung  in  azimuth  until 
the  line  of  sight  falls  on  P,  when  it  is  clamped.  It  is  now 
oriented  for  this  station,  and  pointings  are  taken  on  all  the 
objects  sighted  from  P,  and  on  such  others  as  may  be  sighted 
from  subsequent  stations,  the  alidade  now  revolving  about  the 
point  a on  the  paper.  The  intersections  on  the  plot  of  the 
two  pointings  taken  to  the  same  object  from^  and  P will  evi- 
dently be  the  true  position  on  the  plot  for  those  points  with 
reference  to,  and  to  the  scale  of,  the  line  ap.  These  intersec- 
tions are  shown  in  Fig.  28. 

It  is  evident  that  if  other  points,  as  D or  C,  be  now  occu- 
pied, the  table  oriented  on  either  A or  P,  and  pointings  taken 
on  any  of  the  objects  sighted  from  both  A and  P,  the  third  or 
fourth  line  drawn  to  the  several  objects  should  intersect  the 
first  two  in  a common  point.  This  furnishes  a check  on  the 
work,  and  should  be  taken  for  all  important  points.  It  is  pref- 
erable also  to  have  more  than  two  points  on  the  sheet  pre- 

* It  will  be  noted  that  this  process  of  orienting  the  plane  table  is  practically 
identical  with  that  by  which  the  limb  of  the  transit  is  oriented  in  traversing 
(art.  loi). 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  1 23 


viously  determined.  Thus,  if  B were  also  known  and  plotted 
at  b,  when  the  table  had  been  oriented  on  any  other  station, 
and  a pointing  taken  to  the  fiducial  edge  of  the  ruler 
should  have  passed  through  b. 

As  fast  as  intersections  are  obtained  and  points  located 
the  accompanying  details  should  be  drawn  in  on  the  map  to 
the  proper  scale.  If  distances  are  read  by  means  of  stadia 
wires  on  a rod  held  at  the  various  points  (see  chap.  VIII), 
then  a single  pointing  may  locate  an  object,  the  distance  being 
taken  off  from  a scale  of  equal  parts,  and  the  point  at  once 
plotted  on  the  proper  direction-line.  It  is  now  common  to  do 
this  in  all  plane-table  surveying. 

128.  Location  by  Resection. — This  consists  in  locating 
the  points  occupied  by  pointings  to  known  and  plotted  points. 
The  simple  case  is  where  a single  pointing  has  been  taken  to 
this  point  from  some  known  point,  and  a line  drawn  through 
it  on  the  sheet.  It  is  not  known  what  point  on  this  line 
represents  the  plotted  position  of  this  station.  The  setting  of 
the  instrument  can  therefore  be  but  approximate,  but  near 
enough  for  all  purposes.  The  table  can  be  oriented  as  before, 
there  being  one  pointing  and  corresponding  line  from  a known 
point.  A station  is  then  selected,  a pointing  to  which  is  as 
nearly  90  degrees  from  the  orienting  line  as  possible,  and  the 
alidade  so  placed  that  while  the  telescope  sights  the  object  the 
fiducial  edge  of  the  ruler  passes  through  the  plot  of  the  same 
on  the  sheet.  The  intersection  of  this  edge  with  the  former 
line  to  this  station  gives  the  station’s  true  position  on  the 
sheet.  This  latter  operation  is  called  resection.  Another  re- 
section from  any  other  determined  point  may  be  made  for  a 
check. 

129.  To  find  the  Position  of  an  Unknown  Point  by  Re- 
section on  Three  Known  Points. — This  is  known  as  the 
Three-point  Problem,  and  occurs  also  in  the  use  of  the  sextant 
in  locating  soundings.  It  is  fully  discussed  in  that  connection 


124 


SUJ^VEVING. 


(see  chap.  X.),  so  that  only  a mechanical  solution  suitable 
for  the  problem  in  hand  will  be  given  here.  It  is  under- 
stood there  are  three  known  points,  A,  B,  and  C,  plotted  in 
a,  b,  and  c on  the  map.  The  table  is  set  up  over  any  given 
point  (not  in  the  circumference  of  a circle  through  A,  B,  and 
C\  Fasten  a piece  of  tracing-paper,  or  linen,  on  the  board, 
and  mark  on  it  a point  p for  the  station  /^occupied.  Level 
the  table,  but  of  course  it  cannot  be  oriented.  Take  pointings 
to  A,  B,  and  C,  and  draw  lines  on  the  tracing-paper  from  p 
towards  a,  b,  and  c,  long  enough  to  cover  these  distances  when 
drawn  to  scale.  Remove  the  alidade  and  shift  the  tracing- 
paper  until  the  three  lines  drawn  may  be  made  to  coincide 
exactly  with  the  three  plotted  points  a,  b,  and  c.  The  point 
p is  then  the  true  position  of  this  point  on  the  sheet.  This 
being  pricked  through,  the  table  may  now  be  oriented  and  the 
work  proceed  as  usual. 

130.  To  find  the  Position  of  an  Unknown  Point  by  Re- 
section on  Two  Known  Points. — This  is  called  the  Two- 
point  Problem,  and  but  one  of  several  solutions  will  be  given. 
It  is  evident  that  if  the  table  could  be  properly  oriented  over 
the  required  point,  its  position  on  the  sheet  could  be  at  once 
found  by  resection  on  the  two  known  points.  The  table  may 
be  oriented  in  the  following  manner:  Let  A and  B be  the 
known  points  plotted  in  a and  b on  the  sheet.  Let  C be  the 
unknown  point  whose  position  c on  the  sheet  is  desired. 
Select  a fourth  point  D,  which  may  be  occupied,  and  so  placed 
that  intersections  from  C and  D on  A and  B will  give  angles 
between  30  and  120  degrees.  Fasten  a piece  of  tracing  linen 
or  paper  on  the  board,  marking  a point  at  random.  Set 
up  over  D,  orienting  the  table  as  nearly  as  may  be  by  the 
needle  or  otherwise.  Draw  lines  from  d!  towards  A,  B,  and 
C.  Mark  off  on  the  latter  the  estimated  distance  to  C,  to 
scale,  calling  this  point  C . Set  up  over  C,  with  d over  the 
station,  orienting  on  D by  the  line  c'd\  This  brings  the  table 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  12$ 


parallel  to  its  former  position  at  D,  From  c'  draw  lines  to  A 
and  B,  intersecting  the  corresponding  lines  drawn  from  d in 
a'  and  b' . We  now  have  a quadrilateral  db'c'd  similar  to 
the  quadrilateral  formed  by  the  true  positions  of  the  plotted 
points  abed,  but  it  differs  in  size,  since  the  distance  c'd  was 
assumed,  and  also  in  position  (azimuth),  since  the  table  was 
not  properly  oriented  at  either  station.  Remove  the  alidade, 
and  shift  the  tracing  until  the  line  a'b'  coincides  with  a and  b 
on  the  sheet.  Replace  the  alidade  on  the  tracing,  bringing  it 
into  coincidence  with  dd , c'b\  or  c' d , and  revolve  the  table  on 
its  axis  until  the  line  of  sight  comes  upon  A,  B,  or  D,  as  the 
case  may  be.  The  table  is  now  oriented,  when  the  true  posi- 
tion of  c may  be  readily  found  by  resecting  from  a and  b, 
which,  when  pricked  through,  gives  its  position  on  the  sheet. 

The  student  may  show  how  the  same  result  could  have  been  obtained  with- 
out the  aid  of  tracing-paper. 

If  the  fourth  point  D may  be  taken  in  range  A and  B, 
the  table  may  be  properly  oriented  on  this  range,  and  a line 
drawn  towards  C from  any  point  on  this  range  line  on  the  plot. 
Then  C is  occupied,  and  the  table  again  properly  oriented  by 
this  line  just  drawn,  when  the  true  position  of  c may  be  found 
by  resecting  from  a and  b,  as  before. 

In  general,  if  the  table  can  be  properly  oriented  over  any 
unknown  point  from  which  sights  may  be  taken  to  two  or 
more  known  (plotted)  points,  the  position  of  this  unknown 
point  is  at  once  found  by  resection  from  the  known  points. 

The  student  would  do  well  to  look  upon  the  table  and  the 
attached  plot  as  analogous  to  the  graduated  horizontal  limb  in 
the  transit.  The  principles  and  methods  of  orienting  are  pre- 
cisely similar,  the  pointings  differing  only  in  this,  that  with  the 
transit  the  horizontal  angle,  referred  to  the  meridian,  is  read 
off,  recorded,  and  afterwards  plotted,  while  with  the  plane 
table  this  bearing  is  immediately  drawn  upon  the  sheet. 

131.  The  Measurement  of  Distances  by  Stadia. — This 


126 


SURVEYING. 


method  of  determining  short  distances  is  now  generally  used  in 
connection  with  the  plane  table.  It  is  fully  discussed  in  chap- 
ter VIIL,  where  the  principles  of  its  action  and  its  use  with 
the  transit  are  given  at  length.  The  same  principles,  field 
methods,  and  tables  apply  to  its  use  with  the  plane  table, 
with  such  modifications  as  one  accustomed  to  the  use  of  the 
plane  table  would  readily  introduce.  When  used  in  this  way 
it  enables  a point  to  be  plotted  from  a single  pointing,  it 
being  located  by  polar  coordinates  (azimuth  and  distance),  in- 
stead of  by  intersections. 

EXERCISES  WITH  THE  PLANE  TABLE. 

132.  Make  a plane  table  survey  of  the  college  campus,  measuring  the  length 
of  one  side  for  a base. 

133.  Having  located  several  points  on  the  sheet  by  intersections,  occupy 
them  and  check  their  location  by  resection. 

134.  Locate  a point  (not  plotted)  by  resection  on  three  known  points  (art. 
129). 

135.  Locate  a point  (not  plotted)  by  resection  on  two  known  points,  first 
taking  the  auxiliary  point  D not  in  line  with  AB,  and  then  by  taking  it  in  line 
with  AB.  This  gives  a check  on  the  position  of  the  point,  and  shows  the  ad- 
vantages of  the  second  method  when  it  is  feasible. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  12/ 


CHAPTER  VI. 

ADDITIONAL  INSTRUMENTS  USED  IN  SURVEYING  AND 

PLOTTING. 


THE  ANEROID  BAROMETER. 


136.  The  Aneroid  Barometer  consists  of  a circular  me- 
tallic box,  hermetically  .sealed,  one  side  being  covered  by  a 
corrugated  plate.  The  air  is  mostly  removed,  enough  only 
being  left  in  to  compensate  the  diminished  stiffness  of  the  cor- 


Fig.  29. 


rugated  cover  at  higher  temperatures.  This  cover  rises  or 
falls  as  the  outer  pressure  is  less  or  greater,  and  this  slight 
motion  is  greatly  multiplied  and  transmitted  to  an  index 
pointer  moving  over  a scale  on  the  outer  face.  The  motion 
of  the  index  is  compared  with  a standard  mercurial  barom- 
eter and  the  scale  graduated  accordingly.  Inasmuch  as  all 


128 


SUR  VE  YING. 


barometric  tables  are  prepared  for  mercurial  barometers, 
wherein  the  atmospheric  pressure  is  recorded  in  inches  of 
mercury,  the  aneroid  barometer  is  graduated  so  that  its  read- 
ings are  identical  with  those  of  the  mercurial  column. 

Figure  29  shows  a form  of  the  aneroid  designed  for  eleva- 
tions to  4000  feet  above  or  to  2000  feet  below  sea-level.  It 
has  a vernier  attachment  and  is  read  with  a magnifying-glass 
to  single  feet  of  elevation.  It  must  not  be  supposed,  how- 
ever, that  elevations  can  be  determined  with  anything  like  this 
degree  of  accuracy  by  any  kind  of  barometer.  The  barometer 
simply  indicates  the  pressure  at  the  given  time  and  place,  but 
for  the  same  place  the  pressure  varies  greatly  from  various 
causes.  All  barometric  changes,  therefore,  cannot  be  attrib- 
uted to  a change  in  elevation,  when  the  barometer  is  carried 
about  from  place  to  place. 

If  two  barometers  are  used  simultaneously,  which  have 
been  duly  compared  with  each  other,  one  at  a fixed  point  of 
known  elevation  and  the  other  carried  about  from  point  to 
point  in  the  same  locality,  as  on  a reconnoissance,  then  the 
two  sets  of  readings  will  give  very  close  approximations  to 
the  differences  of  elevation.  If  the  difference  of  elevation  be- 
tween distant  points  is  desired,  then  long  series  of  readings 
should  be  taken  to  eliminate  local  changes  of  pressure.  The 
aneroid  barometer  is  better  adapted  to  surveys  than  the  mer- 
curial, since  it  may  be  transported  and  handled  with  greater 
ease  and  less  danger.  It  is  not  so  absolute  a test  of  pressure, 
however,  and  is  only  used  by  exploring  or  reconnoissance 
parties.  For  fixed  stations  the  mercurial  barometer  is  to  be 
preferred.  It  has  been  found  from  experience  that  the  small 
aneroids  of  i|  to  2\  inches  diameter  give  as  accurate  results 
as  the  larger  ones. 

137.  Barometric  Formulse. — In  the  following  derivation 
of  the  fundamental  barometric  formula  the  calculus  is  used,  so 
that  the  student  will  have  to  take  portions  of  it  on  trust  until 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  1 29 


he  has  studied  that  branch  of  mathematics.  All  that  follows 
Eq.  (4)  he  can  read. 

Let  H — height  of  the  “ homogeneous  atmosphere”*  in  lat. 

45°- 

h — corresponding  height  of  the  mercurial  column, 
d r=  the  relative  density  of  the  “ homogeneous  atmos- 
phere” with  reference  to  mercury. 
z = difference  of  elevation  between  two  points,  with 
barometric  readings  of  h'  and  at  the  higher 
and  at  the  lower  point  respectively. 

Then  from  the  equilibrium  between  the  pressures  of  the 
mercurial  column  and  atmosphere  we  have  : 


h = dH (i) 

Also,  for  a small  change  in  elevation,  dz,  the  corresponding 
change  in  the  height  of  the  mercurial  column  would  be 

dh  =.  6dz (2) 

Substituting  in  (2)  the  value  of  as  given  by  (i),  we  have : 


dh  = -j^dz'y 


or. 


dz  = H- 


h 


Integrating  (3)  between  the  limits  h'  and  we  have: 


, K 
log.  y 


(3) 

(4) 


* “ Homogeneous  atmosphere”  signifies  a purely  imaginary  condition 
wherein  the  atmosphere  is  supposed  to  be  of  uniform  density  from  sea-level  to 
such  upper  limit  as  may  be  necessary  to  give  the  observed  pressure  at  the  ob- 
served temperature. 

9 


130 


SUR  VE  YING. 


where  the  logarithm  is  in  the  Napierian  system.  Dividing  by 
the  modulus  of  the  common  system  to  adapt  it  to  computation 
by  the  ordinary  tables,  we  have : 

2=2.30258// iog„  A (5) 

If  Ho  be  the  height  of  the  homogeneous  atmosphere  at  a 
temperature  of  32°  F.,  and  if  ho  be  the  height  of  the  mercurial 
column  at  sea-level  at  same  temperature,  and  if  and  be 
the  specific  gravities  of  mercury  and  air  respectively,  then, 
evidently, 

or,  Ho  = — — (6) 

From  experiment  we  have  : 

ho  — 29.92  inches, 

.fm=  13-596 

ga  - 0.001239 

whence  Ho  = 26,220  feet. 

This  is  on  the  assumption  that  gravity  is  constant  to  this 
height  above  sea-level.  When  this  is  corrected  for  variable 
gravity  we  have : 

Ho  = 26,284  feet (7) 

Equation  (7)  gives  the  height  of  the  homogeneous  atmos- 
phere at  a temperature  of  32°  F.  But  since  the  volume  of  a gas 
under  constant  pressure  varies  directly  as  the  temperature,  and 
since  the  coefficient  of  expansion  of  air  is  0.002034  for  1°  F., 
we  have  for  the  height  of  the  homogeneous  atmosphere  at  any 
temperature : 


//=//„  [i +0.002034  (/— 32°)]  ...  (8) 


ADJUSTMENT,  USE,  AND  CADE  OF  INSTRUMENTS.  I3I 


If  the  temperature  chosen  be  the  mean  of  the  temperatures  at 
the  two  points  of  observation,  as  t’  and  tx  for  the  upper  and 
lower  points  respectively,  then  we  should  have: 

//=  j^i  + 0.002034  (^--32)] 

= 26,284  [l  +O.OOIOI7  64)]  . . (9) 

Substituting  this  value  of  Hm  Eq.  (5)  we  obtain : 

h 

z — 60,520  [i  + 0.001017  (/'+  — 64)]  log  . (10) 

If  we  wish  to  refer  this  equation  to  approximate  sea-level 
(height  of  mercurial  column  of  30  inches)  and  to  a mean  tem- 
perature of  the  two  stations  of  50°  F.,  we  may  write : 

3? 

, K , h'  , 30  , 30 

K 

Also,  when  t'  lOO®,  we  have 


/'-f /,-64=36". 


Substituting  these  equivalents  in  eq.  (10),  we  obtain 


z = 60520  (i  -|-  0.001017X 


(n) 


8 = 62737  log  ^ - 62737  log 


132 


SUK  VE  YING. 


In  this  equation,  the  two  terms  of  the  second  member  rep- 
resent the  elevations  of  the  upper  and  lower  points  respec- 
tively, above  a plane  corresponding  to  a barometric  pressure  of 
30  inches  and  for  a mean  temperature  of  the  two  positions  of 
50°  F. 

Table  I.  is  computed  from  this  equation,  the  arguments  be- 
ing the  readings  of  the  barometer,  Ji!  and  /;,  at  the  upper  and 
lower  stations  respectively,  the  tabular  results  being  elevations 
above  an  approximate  sea-level.  The  difference  between  the 
two  tabular  results  gives  the  difference  of  elevation  of  the  two 
points,  for  a mean  temperature  of  50°  and  no  allowance  made 
for  the  amount  of  aqueous  vapor  in  the  air.  For  other  tem- 
peratures, and  for  the  effect  of  the  humidity  (which  is  not  ob- 
served, but  the  average  conditions  assumed  to  exist),  a certain 
correction  needs  to  be  applied,  which  correction  is  not  an  abso- 
lute amount,  but  is  always  a certain  proportion  of  the  difference 
of  elevation  as  obtained  from  eq.  (ii)  or  table  I.  If  the  two 
elevations  taken  from  the  table  be  called  A'  and  A^y  and  the 
correction  for  temperature  and  humidity  be  C,  we  would  have 

^ = {A'-A,)(i  + C) (12) 

It  is  seen,  therefore,  that  ^7  is  a coefficient  which,  when  mul- 
tiplied into  the  result  obtained  from  table  L,  gives  the  correc- 
tion to  be  applied  to  that  result.  The  values  of  C are  given 
in  table  II.  for  various  values  of  t'  + 

The  following  example  will  illustrate  the  use  of  the  tables : 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  133 


TABLE  I.  BAROMETRIC  ELEVATION.* 


30 

Containing  A = 62737  log  T • Argument,  h. 

h 


h. 

A. 

Dif.  for 
.01. 

k. 

A. 

Dif.  for 

.CI, 

h. 

A. 

Dif.  for 

.OI« 

Inches. 

Feet. 

Feet. 

Inches. 

Feet. 

Feet. 

Inches. 

Feet. 

Feet. 

II. 0 

27,336 

—24.6 

14.0 

20,765 

-19.5 

17.0 

— 16.0 

II.  I 

27,090 

24.4 

14. 1 

20,570 

19.3 

17. I 

15,316 

15.9 

II  .2 

20,846 

24.2 

14.2 

20,377 

19. 1 

17.2 

15,157 

15.8 

II. 3 

26,604 

24.0 

14.3 

20,186 

18.9 

17.3 

14,999 

15.7 

II. 4 

26,364 

23.8 

14.4 

19,997 

18.8 

17.4 

14,842 

15.6 

II. 5 

26,126 

23.6 

14.5 

19,809 

18.6 

17.5 

14,686 

15.5 

II. 6 

25,890 

23.4 

14.6 

19,623 

18.6 

17.6 

14,531 

15.4 

II. 7 

25,656 

23.2 

14.7 

19,437 

18.5 

17.7 

14,377 

15.4 

II. 8 

25,424 

23.0 

14.8 

19,252 

18.4 

17.8 

14,223 

15.3 

II. 9 

25,194 

22.8 

14.9 

19,068 

18.2 

17.9 

14,070 

15.2 

12.0 

24,966 

22.6 

15.0 

18,886 

18. 1 

18.0 

13,918 

15.1 

12. 1 

24,740 

22.4 

15. 1 

18,705 

18.0 

18. 1 

13,767 

15.0 

12.2 

24,516 

22.2 

15.2 

18,525 

17.9 

18.2 

13,617 

14.9 

12.3 

24,294 

22.1 

15.3 

18,346 

17.8 

18.3 

13,468 

14.9 

12.4 

24,073 

21.9 

15.4 

18,168 

17.6 

18,4 

13,319 

14.7 

12.5 

23,854 

21.7 

15.5 

17,992 

17.5 

18.5 

13,172 

14.7 

12.6 

23,637 

21.6 

15.6 

17,817 

17.4 

18.6 

13,025 

14.6 

12.7 

23,421 

21.4 

15.7 

17,643 

17.3 

18.7 

12,879 

14.6 

12.8 

23,207 

21.2 

15.8 

17,470 

17.2 

18.8 

12,733 

14.4 

12.9 

22,995 

21.0 

15.9 

17,298 

17. 1 

18.9 

12,589 

14.4 

13.0 

22,785 

20.9 

16.0 

17,127 

16.9 

19.0 

12,445 

14.3 

13. 1 

22,576 

20.8 

16. 1 

16,958 

16.9 

19. 1 

12,302 

14.2 

13.2 

22,368 

20.6 

16.2 

16,789 

16.8 

19.2 

12,160 

14.2 

13.3 

22,162 

20.4 

16.3 

16,621 

16.7 

19-3 

12,018 

14.  I 

13.4 

21,958 

20. 1 

16.4 

16,454 

16.6 

19.4 

11,877 

14.0 

13.5 

21,757 

20.0 

16.5 

16,288 

16.4 

19.5 

11,737 

13.9 

13.6 

21,557 

19.9 

16.6 

16,124 

16.3 

19.6 

11,598 

13.9 

13.7 

21,358 

19.8 

16.7 

15  961 

16.3 

19.7 

11,459 

M 

CO 

13.8 

21,160 

19.8 

16.8 

15,798 

16.2 

19.8 

11,321 

13.7 

13.9 

20,962 

-19.7 

16.9 

15,636 

— 16.0 

19.9 

11,184 

-13.7 

14.0 

20.765 

17.0 

15,476 

20.0 

11,047 

* This  table  taken  from  Appendix  10,  Report  U.  S.  Coast  and  Geodetic 
Survey,  i88i. 


134 


SURVEYING. 


TABLE  I.  Barometric  Elevation. — Continued. 


30 

Containing  A = 62737  log  — . Argument,  h. 


h. 

A. 

Dif.  for 
.01. 

h. 

A. 

Dif.  for 
.01, 

h. 

A. 

Dif.  for 
.01. 

Inches. 

Feet. 

Feet. 

Inches. 

Feet. 

Feet. 

Inches. 

Feet. 

Feet. 

20.0 

11,047 

— 13.6 

23.0 

7,239 

-II. 8 

26.0 

3,899 

— 10.5 

20. 1 

10,911 

13-5 

23.  I 

7,121 

II. 7 

26.  I 

3.794 

10.4 

20.2 

10,776 

13.4 

23.2 

7,004 

II. 7 

26.2 

3,690 

10.4 

20.3 

10,642 

13.4 

23.3 

6,887 

II. 7 

26.3 

3,586 

10.3 

20.4 

10,508 

13-3 

23-4 

6,770 

II  .6 

26.4 

3,483 

10.3 

20.5 

10,375 

13-3 

23-5 

6,654 

II  .6 

26.5 

3.380 

10.3 

20.6 

10,242 

13-2 

23.6 

6,538 

II-5 

26.6 

3,277 

10.2 

20.7 

10,110 

I3-I 

23.7 

6,423 

II. 5 

26.7 

3,175 

10.2 

20.8 

9.979 

I3-I 

23.8 

6,308 

II. 4 

26.8 

3,073 

10.  I 

20.9 

9,848 

13.0 

23-9 

6,194 

II. 4 

26.9 

2.972 

10.  I 

j 21.0 

9,718 

12.9 

24.0 

6,080 

11-3 

27.0 

2,871 

10. 1 

21. I 

9.589 

12.9 

24.1 

5,967 

II. 3 

27.1 

2,770 

10. 0 

21.2 

9,460 

12.8 

24.2 

5.854 

1 

II. 3 

27.2 

2,670 

10. 0 

21.3 

9.332 

12.8 

24.3 

5,741 

II. 2 

27.3 

2,570 

10. 0 

21.4 

9,204 

12.7 

24.4 

5.629 

ii.  I 

27.4 

2,470 

9.9 

21.5 

9.077 

12.6 

24.5 

5,518 

II.  I 

27.5 

9.9 

21.6 

8,951 

12.6 

24.6 

5,407 

II . I 

27.6 

2,272 

9.9 

21.7 

8,825 

12.5 

24.7 

5.296 

II. 0 

27.7 

2,173 

9.8 

21.8 

8,700 

12.5 

24.8 

5,186 

10.9 

27.8 

2,075 

CO 

O' 

21.9 

8,575 

12.4 

24.9 

5,077 

10.9 

27.9 

1,977 

9-7 

22.0 

8,451 

12.4 

25.0 

4,968 

10.9 

28.0 

1,880 

9.7 

22.1 

8,327 

12.3 

25.1 

4,859 

10.8 

28.1 

1,783 

9.7 

22.2 

8,204 

12.2 

25.2 

4,751 

10.8 

28.2 

1,686 

9.7 

22.3 

8,082 

12.2 

25.3 

4,643 

10.8 

28.3 

1,589 

9.6 

22.4 

7,960 

12.2 

25-4 

4,535 

10.7 

28.4 

1,493 

9.6 

22.5 

7,838 

12. 1 

25.5 

4,428 

10.7 

28.5 

1,397 

9.5 

22.6 

7,717 

12.0 

25.6 

4,321 

10.6 

28.6 

1,302 

9.5 

22.7 

7,597 

12.0 

25.7 

4,215 

10.6 

28.7 

1,207 

9.5 

22.8 

7,477 

II. 9 

25.8 

4,109 

10.5 

28.8 

1,112 

9.4 

22.9 

7,358 

-II. 9 

25-9 

4,004 

-10.5 

28.9 

1,018 

-9.4 

23.0 

7.239 

26.0 

3,899 

29.0 

924 

ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  135 


TABLE  I.  Barometric  Elevations. — Continued. 


30 

Containing  A = 62737  log  ~j^-  Argument,  h. 


\h. 

A. 

Dif.  for 
.01. 

h. 

A. 

Dif.  for 

• OI, 

h. 

A. 

Dif.  for 
.01. 

Inches. 

29.0 

29.1 

29.2 

29*3 

29.4 

29-5 

29.6 

29.7 

Feet. 

924 

830 

736 

643 

550 

458 

366 

274 

Feet. 

-9.4 

9.4 

9-3 

9-3 

9.2 

9.2 

-9.2 

Inches. 

29.7 

29.8 

29.9 

30.0 

30.1 

30.2 

30.3 

30.4 

Feet. 

274 

182 

91 

00 

-91 

181 

271 

361 

Feet. 

-9.2 

9.1 

9.1 

9.1 

9.0 

9.0 

-9.0 

Inches. 

30.4 

30.5 

30.6 

30-1 

30.8 

30.9 

31.0 

Feet. 

361 

451 

540 

629 

717 

805 

— 893 

Feet. 

—9.0 

8.9 

8.9 

8.8 

'8.8 

-8.8 

TABLE  11. 

CORRECTION  COEFFICIENTS  TO  BAROMETRIC  ELEVATIONS 
FOR  TEMPERATURE  AND  HUMIDITY.* 


tx  + 1>. 

c. 

tx  + t'. 

c. 

tx  + t'. 

c. 

0° 

—0. 1025 

60 

— C 

>.0380 

120 

-I-0.0262 

5 

— .0970 

65 

- 

.0326 

125 

+ -0315 

10 

— .0915 

70 

— 

.0273 

130 

+ .0368 

15 

— .0860 

75 

- 

.0220 

135 

+ .0420 

20 

— .0806 

80 

- 

.0166 

140 

+ -0472 

25 

- .0752 

85 

- 

.0112 

145 

+ -0524 

30 

— .0698 

90 

- 

.0058 

150 

+ -0575 

35 

- .0645 

95 

- 

.0004 

155 

-{-  .0626 

40 

- .0592 

100 

+ 

.0049 

160 

+ -0677 

45 

- -0539 

105 

+ 

.0102 

165 

+ .0728 

50 

— .0486 

ITO 

+ 

.0156 

170 

+ .0779 

55 

- .0433 

II5 

+ 

.0209 

175 

-f-  .0829 

60 

— .0380 

120 

+ 

.0262 

180 

+ .0879 

*This  table  compiled  from  tables  I.  and  IV.  of  Appendix  10  of  Report  of 
the  U.  S.  Coast  and  Geodetic  Survey  for  1881. 


136 


SU/^  VE  YING. 


Example, 

From  observations  made  at  Sacramento,  Cal.,  and  at  vSum- 
mit  on  the  top  of  the  Sierra  Nevada  Mountains,  the  annual 
means  were : 


//  = 23.288  in.  /'  = 42.i/^ 
— 30.014  in.  — 59.9. 

From  table  I.  we  have 

A'  = 6901.0  feet. 
A^=  — 12.7  “ 

^'-^,  = 6913.7  “ 


From  table  II.  we  find  for  = i02°.o,  C = .0070. 

. • . .S'  = 6913.7  (i  + .0070)  = 6962  feet. 

138.  Use  of  the  Aneroid. — The  aneroid  barometer  should 
be  carried  in  a leather  case,  and  it  should  not  be  removed  from 
it.  It  should  be  protected  from  sudden  changes  of  tempera- 
ture, and  when  observations  are  made  it  should  have  the 
temperature  of  the  surrounding  outer  aif  It  should  not  be 
carried  so  as  to  be  affected  by  the  heat  of  the  body,  and  should 
be  read  out  of  doors,  or  at  least  away  from  all  artificially 
warmed  rooms.  Always  read  it  in  a horizontal  position.  The 
index  should  be  adjusted  by  means  of  a screw  at  its  back,  to 
agree  with  a standard  mercurial  barometer,  and  then  this  ad- 
justment left  untouched. 

When  but  a single  instrument  is  used  it  is  advisable  to  pass 
between  stations  as  rapidly  as  possible,  but  to  stop  at  a number 
of  stations  during  the  day  for  a half-hour  or  so,  reading  the 
barometer  on  arrival  and  on  leaving.  The  difference  of  these 
two  readings  shows  the  rate  of  change  of  barometric  readings 
due  to  changing  atmospheric  conditions,  and  from  these  iso- 
lated rates  of  change  a continuous  correction-ctirve  can  be  con- 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  13/ 


structed  on  profile  or  cross-section  paper  from  which  the 
instrumental  corrections  can  be  taken  for  any  hour  of  the 
day.*  The  observations  should  be  repeated  the  same  day  in 
reverse  order,  the  corrections  applied  as  obtained  from  this 
correction  curve,  and  the  means  taken.  Observations  should 
be  made  when  the  humidity  of  the  air  is  as  nearly  constant  as 
possible,  and  never  in  times  of  changeable  or  snowy  weather. 

Let  the  student  measure  the  heights  of  buildings,  hills,  etc., 
and  then  test  his  results  by  level  or  transit. 

THE  PEDOMETER. 

139.  The  Pedometer  is  a pocket-instrument  for  register- 
ing the  number  of  paces  taken  when  walking.  It  is  generally 


Fig.  30,— Front  View. 


Fig.  31.— Back  View. 


made  in  the  form  of  a watch,  the  front  and  back  views  being 
shown  in  Figs.  30  and  31. 


* Mr.  Chas.  A.  Ashburner,  Geologist  of  the  Penn.  Geol.  Survey,  has  used 
this  method  with  good  results. 


138 


SURVEYING. 


When  the  instrument  is  attached  to  the  belt  in  an  upright 
position,  as  here  shown,  the  jar  given  it  at  each  step  causes  tlie 
weighted  lever  shown  in  Fig.  31  to  drop  upon  the  adjustable 
screw  vS.  The  lever  recovers  its  position  by  the  aid  of  a spring, 
and  in  so  doing  turns  a ratchet-wheel  by  an  amount  propor- 
tional to  the  amplitude  of  the  lever’s  motion.  T.his  may  be 
adjusted  to  any  length  of  pace  by  means  of  the  screw  5,  which 
is  turned  by  a key.  The  face  is  graduated  like  that  of  a watch, 
and  gives  the  distance  travelled  in  miles.  This  instrument 
may  also  be  used  on  a horse,  and  when  adjusted  to  the  length 
of  a horse’s  step  will  give  equally  good  results.  The  accuracy 
of  the  result  is  in  proportion  to  the  uniformity  of  the  steps, 
after  having  been  adjusted  properly  for  a given  individual. 
The  instrument  is  only  used  on  explorations,  preliminary  sur- 
veys, and  reconnoissance-work. 

The  Length  of  Mens  Steps  has  been  investigated  by  Prof. 
Jordan,*  of  the  Hanover  Polytechnic  School.  From  256 
step-measurements  by  as  many  different  individuals,  of  lines 
from  650  to  1000  feet  in*  length,  carefully  measured  by 
rods  and  steel  tapes,  he  concludes  that  the  average  length  of 
step  is  2.648  feet,  ranging  from  2.066  to  3.182  feet.  The  mean 
deviation  from  this  amount  for  a single  measurement  was 
± 0.147  feet,  or  5^  per  cent.  The  average  age  of  the  persons 
making  these  step-measurements  was  20  years.  The  length  of 
step  decreases  with  the  age  of  the  individual  after  the  age  of 
25  to  30  years.  It  is  also  proportional  to  the  height  of  the 
person.  The  results  for  18  different-sized  persons  gave  the 
following  averages : 


Height  of  person 

5'.o8 

5'.25 

5'4i 

5'.58 

5'-74 

5'-9f> 

6'.07 

6'. 23 

6 '.40 

6'.56 

Length  of  step. . 

2 .46 

2 .53 

2 .56 

2 .59 

2 .62 

2 .69 

2 .72 

2 .76 

2.79 

2.85 

* See  translation  in  Engineering  News  and  American  Contract  Journal  for 
July  25,  1885. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  1 39 


On  slopes  the  step  is  always  shorter  than  on  level  ground, 
whether  one  goes  up  or  down.  The  following  averages  from 
the  step-measurement  of  136  lines  on  mountain-slopes  along 
trails  were  found : 


Slope 

0° 

5° 

10° 

15° 

20° 

25° 

30° 

Length  of  step  in  ascending 

2'.53 

2'. 30 

2'.03 

i'.84 

i'.64 

i'48 

l'.25 

Length  of  step  in  descending 

2'.53 

2'.43 

2'. 36 

2'.30 

2'. 20 

T.97 

I '.64 

The  length  of  the  step  is  also  found  to  increase  with  the 
length  of  the  foot.  One  steps  farther  when  fresh  than  when 
tired.  The  increase  in  the  length  of  the  step  is  also  in  nearly 
direct  proportion  to  the  increase  of  speed  in  walking. 

When  the  proper  personal  constants  are  determined,  and 
when  walking  at  a constant  rate,  distances  can  be  determined 
by  pedometer,  or  by  counting  the  paces,  to  within  about  two 
per  cent  of  the  truth.  One  should  always  take  his  7iatural  step, 
and  not  an  artificial  one  which  is  supposed  to  have  a known 
value,  as  three  feet,  for  instance.  Let  a base  be  measured  off 
and  each  student  determine  the  length  of  his  natural  step  when 
walking  at  his  usual  rate,  or,  what  is  the  same  thing,  find  how 
many  paces  he  makes  in  icx)  feet.  He  then  has  always  a 
ready  means  of  determining  distances  to  an  approximation, 
which  in  many  kinds  of  work  is  abundantly  sufficient. 


THE  ODOMETER. 

140.  The  Odometer  is  an  instrument  to  be  attached  to 
the  wheel  of  a vehicle  to  record  the  number  of  revolutions 
made  by  it.  One  form  of  such  an  instrument  is  shown  in 
Fig.  32  attached  to  the  spokes  of  a wheel. 

Each  revolution  is  recorded  by  means  of  the  revolution  of 
an  axis  with  reference  to  the  instrument,  this  axis  really  being 


140 


SURVEYING. 


held  stationary  by  means  of  an  attached  pendulum  which  does 
not  revolve.  The  instrument  really  revolves  about  this  fixed 
axis  at  each  revolution  of  the  wheel,  and  the  number  of  times 


Fig.  32. 


it  does  this  is  properly  recorded  and  indicated  by  appropriate 
gearing  and  dials. 

This  method  of  measuring  distances  is  more  accurate  than 
by  pacing,  as  the  length  of  the  circumference  of  the  wheel  is  a 
constant.  This  length  multiplied  by  the  number  of  revolu- 
tions is  the  distance  travelled.  It  is  mostly  used  by  exploring 
parties  and  in  military  movements  in  new  countries  which  have 
not  been  surveyed  and  mapped. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  I4I 


THE  CLINOMETER. 

141.  The  Clinometer  is  a hand-instrument  for  determin- 
ing the  slope  of  ground  or  the  angle  it  makes  with  the  horizon. 
It  consists  essentially  of  a level  bubble,  a graduated  arc,  and  a 
line  of  sight,  so  joined  that  when  the  line  of  sight  is  at  any  angle 
to  the  horizon  the  bubble  may  be  brought  to  a central  position 
and  the  slope  read  off  on  the  graduated  arc.  Such  a combina- 
tion is  shown  in  Fig.  33.  It  is  called  the  Abney  level  and 


clinometer,  being  really  a hand-level  when  the  vernier  is  set  to 
read  zero.  The  position  of  the  bubble  is  visible  when  looking 
through  the  telescope,  the  same  as  with  the  Locke  hand-level, 
shown  in  Fig.  16,  p.  82.  The  body  of  the  tube  is  made  square, 
so  that  it  may  be  used  to  find  vertical  angles  of  any  surface  by 
placing  the  tube  upon  it  and  bringing  the  bubble  to  the  centre. 
The  graduations  on  the  inner  edge  of  the  limb  give  the  slope 
in  terms  of  the  relative  horizontal  and  vertical  components  of 
any  portion  of  the  line;  thus,  a slope  of  2 to  i signifies  that 
the  horizontal  component  is  twice  the  vertical.  In  reading  this 
scale  the  edge  of  the  vernier-arm  is  brought  into  coincidence 
with  the  graduation. 

This  instrument  is  very  useful  in  giving  approximate  slopes 
in  preliminary  surveys,  the  instrument  being  pointed  to  a posi- 


142 


SUJiVEYING. 


tion  as  high  above  the  ground  as  its  own  elevation  when  held 
to  the  eye. 

THE  OPTICAL  SQUARE. 

142.  The  Optical  Square  is  a small  hand-instrument  used 
to  set  off  a right  angle.  It  is  shown  in  Fig.  34,  the  method  of 


its  use  being  evident  from  the  figure.  Thus,  while  the  rod  at 
0 is  seen  directly  through  the  opening,  the  rod  at  p is  seen  in 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  I43 


the  glass  as  the  prolongation  downwards  of  that  of  Oy  it  being 
reflected  from  the  mirrors  /and  ^ in  succession,  they  having 
an  angle  of  45°  with  each  other.  By  this  means  a line  may  be 
located  at  right  angles  to  a given  line  at  a given  point,  or  a 
point  in  a given  line  may  be  found  in  the  perpendicular  to  this 
line  from  a given  point. 

THE  PLANIMETER. 

143.  The  Planimeter  is  an  ingenious  instrument  used  for 
measuring  irregular  areas.  It  is  a marked  example  of  high 
mathematical  analysis  embodied  in  a very  simple  and  useful 
mechanical  appliance.  Many  forms  of  it  are  now  in  use,  three 


of  the  best  of  which  will  be  described.  The  instrument  has 
come  to  be  a necessity  in  all  kinds  of  surveying  and  engineer- 
ing work  where  irregular  areas  have  to  be  evaluated.  It  is 
important  that  the  student  should  thoroughly  understand  its 
principles,  that  he  may  use  it  with  the  greatest  efficiency. 
The  demonstration  of  its  competency  to  measure  areas  is 
necessarily  somewhat  involved,  and  requires  a little  patient 
consideration.  The  demonstrations  here  given,  though  fol- 
lowing the  methods  of  the  calculus,  are  free  from  the  peculiar 
notation  there  used.  The  form  of  the  instrument  shown  in 
Fig.  35  is  known  as  Amsler’s  Polar  Planimeter.  The  point  e 
is  fixed  by  means  of  a needle-point  puncturing  the  paper.  The 
point  d is  made  to  pass  over  the  perimeter  of  the  area  to  be 
measured,  and  the  record  given  by  the  rolling-wheel  c and  the 


144 


SUR  VE  YING. 


record-disk  / is  the  area  in  the  unit  for  which  the  length  of  the 
arm  h was  set.  The  rolling-wheel  is  mounted  on  an  axis 
which  is  parallel  to  the  arm  h,  and  moves  with  a minimum 
amount  of  friction.  It  is  evident  that  any  motion  of  the 
wheel  c in  the  direction  of  its  axis  would  not  cause  it  to  re- 
volve, while  any  motion  at  right  angles  to  this  axis  is  fully 
recorded  by  the  wheel.  The  arm  ei  is  of  fixed  length,  while 
the  length  of  the  arm  h is  adjustable. 

144.  Theory  of  the  Polar  Planimeter.* — In  Fig.  36  let  C 
represent  the  point  where  the  instrument  is  fastened  to  the 
paper,  and  ClPthe  arm,  of  fixed  length  m,  whose  only  motion 


is  that  of  revolution  on  ^7  as  a centre,  causing  P to  move  in  a 
circular  arc.  RT  the  other  arm,  revolving  on  P as  a centre, 
and  carrying  at  the  fixed  distance  RP  n)  from  Pa  rolling- 
wheel  whose  periphery  touches  the  paper  at  R and  whose  axis 
is  parallel  to  PP.  RT  also  carries  at  a distance  TP  1)  from 
P the  tracing-point,  P ; / is  a constant  while  the  instrument  is 
in  use,  though  capable  in  the  best  instruments  of  having  dif- 
ferent values  given  to  it  for  different  purposes. 


* The  demonstration  here  given  was  published  by  Mr.  Fred.  Brooks,  in  the 
Journal  of  the  Association  of  Engineering  Societies,  vol.  iii.,  p.  294,  and  is  rep- 
resented as  a joint  production  by  himself  and  Mr.  Frank  S.  Hart.  A few  slight 
changes  and  additions  are  here  made. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  I45 


T and  R can  move  nearer  to  or  further  from  C only  by  the 
motion  of  the  arm  TR  on  Z’  as  a centre  varying  the  angle  X. 

The  distance  CT  = + / cos  X)'  + (/  sin  XJ 

= ^ R 2rnl  cos 

as  may  be  seen  by  dropping  a perpendicular  Tq  from  T on  CP 
produced. 

To  every  particular  value  of  X correspond  particular  values 
of  CT,  CR,  angle  CRP,  etc. ; and  successive  small  variations  in 
X are  accompanied  by  successive  small  variations  in  these 
quantities.  When  T,  starting  at  any  given  distance  from  C,  is 
moved  through  any  path  to  the  same  or  another  place  equally 
distant  from  C,  the  usefulness  of  the  instrument  depends  upon 
W’s  coming  back  to  its  first  value  by  passing  in  reverse  order 
through  the  changes  it  has  once  made.  This  is  secured  by 
the  usual  construction  of  the  instrument,  which  prevents  T 
and  R from  crossing  the  line  of  CP',  in  other  words,  X,  ex- 
pressed as  arc  to  radius  unity,  is  never  less  than  0 nor  more 
than  7t  (a  half-circumference). 

The  only  motion  possible  besides  those  above  described  is 
the  turning  of  the  rolling-wheel  on  its  axis,  which  is  produced 
by  the  component  of  the  motion  of  R perpendicular  to  RP, 
that  is,  tangential  to  its  periphery ; but  the  wheel  does  not 
turn  for  the  component  of  the  motion  of  R in  the  direction 
RP,  which  is  parallel  to  the  axis.  Suppose,  for  simplicity,  that 
the  periphery  of  the  wheel  has  a length  of  one  unit  and  that 
the  number  of  turns  and  fractions  of  a turn  it  makes  upon  any 
trial  is  recorded ; for,  whatever  the  size  and  graduations  may 
be,  a simple  calculation  would  reduce  the  results  to  the  re- 
quired equivalents.  To  illustrate,  let  the  arm  RT  turn  on  P 
as  a centre,  while  CP  remains  fixed,  from  the  position  of  the 
full  line  to  that  of  the  dotted  line  sk ; R moves  to  s,  describing 
10 


146 


SUR  VE  YING. 


an  arc  which  is  everywhere  at  right  angles  to  its  radius  RP ; 
hence  the  record  of  the  wheel  is  the  length  of  the  arc  Rs.  On 

the  other  hand,  supposing  that  CsP  is  a right  angle  -j  and 

11 

cos  CPs  ~ — , let  both  arms  revolve  around  C with  X fixed 
m 

equal  to  CPs ; the  wheel  is  at  every  point  moving  parallel  to 
its  own  axis,  and  its  record  is  zero.  The  distance  Ck  of  the 
tracing-point  from  C in  this  case  may  be  found  by  substituting 
n 

the  value  — for  the  cos  X in  the  general  expression  for  CT, 

which  gives  V nf  P 2nl.  The  circumference  described  by 
the  tracing-point  with  this  radius  may  be  called  the  zero-cir- 
cumference. 

If  both  arms  similarly  revolved  around  C with  X fixed  at 
any  other  value  between  0 and  n,  the  axis  of  the  wheel  would 
make  an  oblique  angle  with  the  direction  of  i?’s  path,  and  the 
wheel  would  partly  roll  and  partly  slip.  The  further  CRP 

7t 

varied  from  — , the  less  in  proportion  would  be  the  slipping 
component,  and  the  greater  the  rolling  component  and  the 

7t 

record  of  the  wheel.  With  CRP'>  T would  describe  an 

arc  outside  the  zero-circumference  and  the  wheel  would  make 

n 

what  we  will  call  a positive  record.  With  CRP  < T would 

describe  an  arc  inside  the  zero-circumference  and  the  wheel 
would  turn  in  the  contrary  direction,  which  we  will  call  nega- 
tive ; provided  that  T revolved  in  the  same  direction  in  both 
cases.  Motion  of  T through  any  arc  in  the  direction  of  the 
hands  of  a watch  may  be  considered  positive ; then  motion  of 
T in  the  opposite  direction  over  the  same  arc  back  to  its  start- 
ing-place must  be  considered  negative,  and  would  obviously  be 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  I47 


attended  by  a turning  of  the  wheel  equal  in  amount  to  that 
attending  the  positive  movement,  but  with  its  direction  re- 
versed. 

In  the  practical  use  of  the  instrument  T may  move  over 
any  path,  near  enough  to  the  zero-circumference  to  be  reached, 
whose  beginning  and  end  are  equally  distant  from  C.  Hence 
X is  the  same  at  the  end  as  at  the  beginning.  The  record 
thus  made  on  the  wheel  is  proportional  to  the  area  included 
between  the  zero-circumference  and  T\  path  and  the  radial 
lines  through  its  beginning  and  end  from  the  centre  as  will 
now  be  explained. 

7"’s  path  may  be  resolved  into  an  infinite  number  of  parts, 
consisting  of  infinitesimal  arcs  (/)  described  from  Pas  a centre 
by  changes  in  X,  CP  being  fixed,  and  of  infinitesimal  arcs  (y) 
described  from  (7  as  a centre  with  X fixed.  This  is  illustrated 
by  large  arcs  of  the  two  classes  on  the  diagram.  The  area  in 
question  may  be  correspondingly  divided  into  elementary  por- 
tions (illustrated  by  the  large  divisions  made  on  the  diagram 
by  fine  radial  lines)  each  of  which  may  be  described  as  plus 
the  area  included  between  one  of  these  infinitesimal  arcs  and 
radial  lines  through  its  extremities  from  (7,  minus  the  sector 
included  by  the  same  radii  and  an  arc  of  the  zero-circumfer- 
ence. Hence  the  area  is  a minus  quantity  if  T moves  inside 
the  zero-circumference,  positive  if  outside ; provided  that  T 
moves  around  C in  the  direction  of  the  hands  of  a watch.  If 
T moves  around  (7  in  a contrary  direction,  both  the  signs  in 
the  above  expression  are  to  be  changed ; for  as  the  area  of  a 
sector  is  equal  to  its  arc  multiplied  by  half  its  radius,  the  area 
becomes  negative  when  the  arc  becomes  negative.  If  this  be 
borne  in  mind  it  will  be  seen  that  the  algebraic  sum  of  all  the 
elements  corresponding  to  the  second  term  in  the  above  ex- 
pression is  the  sector  of  the  zero-circumference  included  by 
radii  passing  through  the  points  of  beginning  and  ending  of 


148 


SURVEYING. 


T's  path ; and  that  the  algebraic  sum  of  all  the  elements 
corresponding  to  the  first  term  is  the  area  inclosed  by  T's 
path  and  lines  from  C to  its  beginning  and  end,  however  irreg- 
ular T's  path  may  be. 

We  will  first  consider  that  class  of  infinitesimal  arcs  (/)  and 
corresponding  elements  of  area,  due  to  changes  in  X alone. 
Their  accumulated  effect  upon  both  the  area  and  up07i  the 
record  of  the  rolling-wheel  is  zero.  As  to  the  wheel,  from  the 
condition  that  X passes  again  in  reverse  order  through  the 
changes  it  has  once  made,  it  follows  that  for  every  infinitesi- 
mal motion,  like  Rs,  of  R,  recorded  by  the  wheel  for  the 
infinitesimal  change  (/)  between  two  consecutive  values  of  W, 
there  must  be  in  some  other  place  a motion  in  the  opposite 
direction  of  the  same  magnitude  for  the  infinitesimal  change 
back  again  between  two  consecutive  values  of  X equal  to  the 
former  pair.  As  to  the  area,  each  infinitesimal  arc  / (like  Tk) 
has,  as  previously  stated,  its  corresponding  element  of  area  ; 
and  the  equally  large  arc  with  the  contrary  sign,  just  now 
referred  to,  in  another  place  where  X has  the  same  values, 
must  also  have  its  corresponding  element  of  area,  exactly  as 
large  as  the  former,  but  with  its  algebraic  sign  reversed.  The 
effect  of  the  first  class  of  elements  into  which  T's  path  was 
resolved  is  thus  eliminated. 

Hence  the  total  record  of  the  wheel  for  T's  whole  path  is 
the  record  due  to  the  second  class  of  its  elements,  the  infini- 
tesimal arcs  (y)  described  from  C with  X fixed  for  each ; and 
the  total  area  included  between  the  zero-circumference,  T's 
path,  and  the  terminal  radii  is  the  sum  of  all  the  elements  of 
area  corresponding  to  this  second  class  of  arcs  which  we 
have  now  to  consider.  J expresses  in  terms  of  arc  to  radius 
unity  any  infinitesimal  angle  ydy  between  radial  lines  passing 
from  C through  the  extremities  of  an  infinitesimal  arc  Tf. 
The  corresponding  element  of  area  is  the  difference  between 
the  sector  TfC  and  the  sector  included  by  the  zero-circumfer- 


ADJUSTMENT,  USE,  AND  CADE  OF  INSTRUMENTS.  I49 


ence  and  the  same  radii.  Making  use  of  the  algebraic  expres- 
sions given  above, 

from  \J  (11^  + 2^/  cos.  X) 

subtract  J + 2?^/) 

and  the  difference  J I {in  cos.  X — n) 

is  the  required  element  of  area. 

The  corresponding  record  of  the  wheel  is  made  by  the 
motion  of  R through  the  path  Re  = / X CR,  This  path  may 
be  resolved  into  two  components,  Rh,  which  has  no  effect 
upon  the  record,  and  he,  which  is  the  record  — J CR  X cos 
{n  — CRP).  By  dropping  the  perpendicular  Cg  upon  PR 
produced  it  will  be  seen  that  CR  cos  {rt  --  CRP)  = Rg  = 
m cos  X — 11.  Hence  record  of  wheel  is  y X {m  cos  X — n). 
Therefore  the  element  of  area  corresponding  to  an  infinitesimal 
arc,J,  is  just  I times  the  record  due  to  the  same  arc  ; hence  the 
sum  of  the  elements  of  area  for  all  the  arcs  {f')  is  / times  the 
total  record  corresponding,  which  is  the  essential  thing  that 
was  to  be  proved. 

In  the  application  of  the  instrument  to  get  the  area  of  a 
closed  figure,  7"’s  path  ends  in  the  same  point  where  it  began, 
and  we  have  two  cases  according  as  this  is  accomplished  by 
67^’s  making  a complete  revolution  around  C,  or  by  its  mov- 
ing backward  as  much  as  it  has  once  moved  forward.  In 
the  first  case,  C is  within  the  figure ; in  the  second,  outside. 
In  both  cases  the  area  between  7”s  path  and  the  terminal 
radii  is  the  area  of  the  closed  figure.  The  sector  within  the 
zero-circumference,  which  we  have  been  deducting,  is  in  the 
first  case  the  whole  circle  n {ml  + + 2nl) ; in  the  second, 

nothing.  Hence  add  ir  {m^  + + 2nl)  to  / times  the  record 

in  the  first  case,  and  add  nothing  to  it  in  the  second,  in  order 
to  get  the  required  area  of  the  closed  figure. 


150 


SUR  VE  YING. 


To  show  that  the  proper  summation  is  made  on  the  wheel 
for  the  areas  outside  and  inside  the  zero-circle, 

let  Af,  — area  generated  by  the  line  CT  when  the  point 
T is  outside  of  circle  ; 

“ Ai  = area  generated  by  the  line  CT  when  the  point 
T is  inside  of  circle  ; 

“ 5 = area  of  sector  between  radii  to  points  where 

the  perimeter  crosses  the  zero-circle  ; 

“ A — area  of  the  figure. 

Then  Ao  — S — outer  area,  and 
S — Ai  = inner  area. 

The  sum  of  these  is 

A = (A,-S)  + {S-A,)  = A„-yj,. 

But  since  Ai  is  recorded  negatively  on  the  wheel,  while  Ao 
is  recorded  positively,  the  wheel  record  is 

Ao— { — Ai)  = Ao-^  Ai  = A. 

145.  To  find  Length  of  Arm  to  give  area  in  any  desired 
unit.  In  the  previous  article  it  was  shown  that  the  area  was 
always  / times  the  wheel  record,  where  / was  the  length  of  the 
arm  carrying  the  tracing-point,  or  the  distance  PT  in  Fig.  36. 
The  wheel  record  is  evidently  its  net  circumferential  move- 
ment, or  nc,  where 

n = number  of  revolutions  of  wheel  shown  by  the  differ- 
ence between  the  initial  and  final  readings, 
and  c = circumference  of  wheel. 

We  may  then  write  for  the  area  of  the  figure 


A = Inc. 


ADJUSTMENT,  USE,  AND  CARE  OE  INSTRUMENTS.  15I 


If  I and  c are  given  in  inches  A will  be  found  in  square  inches, 
and  the  same  for  any  other  unit.  To  cause  an  area  of  i square 
inch  to  register  i revolution  of  the  wheel,  we  will  have 


I = Ic, 


If  c were  2 inches,  this  would  give  l—\  inch,  which  would 
be  too  short  for  practical  purposes.  Let  us  assume,  then,  that 
I square  inch  shall  be  registered  as  o.i  revolution  of  the 
wheel.  Then  we  have 


I = O.I  Ic, 


On  an  instrument  the  author  has  used  c — 2.347  inches, 
whence  for  0.1  revolution  to  correspond  to  i square  inch  area 
we  have 

/ = = 4.26  inches. 

2.347 

When  this  length  of  arm  is  carefully  set  off  by  the  appro- 
priate clamp-  and  slow-motion  screw,  the  area  is  given  in 
square  inches  by  multiplying  the  number  of  revolutions  of  the 
wheel  by  10.  A vernier  is  provided  for  reading  the  revolu- 
tions of  the  wheel  to  thousandths ; hence  if  it  be  read  to 
thousandths,  and  two  figures  pointed  off,  the  result  is  the  area 
of  the  diagram  moved  over  in  square  inches. 

It  is  evident  that  c can  be  evaluated  in  centimetres,  and  the 
corresponding  metrical  length  of  / found  for  giving  the  result 
in  the  metric  notation.  The  exact  circumference  of  the  wheel 
is  determined  by  the  makers,  and  remains  a constant  for  that 


152 


SUR  VE  YING. 


individual  instrument,  giving  a corresponding  set  of  values 
of  /.  Since  no  two  instruments  are  likely  to  have  exactly  the 
same  wheel-circumference,  so  the  settings  for  one  instrument 
cannot  be  used  for  another. 

It  must  be  kept  in  mind  that  the  result  is  given  in  absolute 
units  of  area  of  the  diagrani,  and  this  result  must  then  be 
evaluated  according  to  the  significance  of  such  unit  on  the 
diagram.  Thus,  if  a sectional  area  has  been  plotted  with  a* 
vertical  scale  of  lo  feet  to  the  inch  and  a horizontal  scale  of 
100  feet  to  the  inch,  then  one  square  inch  on  this  diagram  rep- 
resents looo  square  feet  of  actual  sectional  area.  The  number 
of  square  inches  in  the  figure  as  given  by  the  planimeter  must 
then  be  multiplied  by  lOOO  to  give  the  area  of  the  section  in 
square  feet. 

146.  The  Suspended  Planimeter.— This  is  shown  in  Fig. 
37.  It  is  essentially  a polar  planimeter,  the  pole  being  at  C, 


Fig.  37. 


It  has  the  advantage  of  allowing  the  wheel  to  move  over  the 
smooth  surface  of  the  plate  5,  instead  of  over  the  paper,  thus 
giving  an  error  about  one  sixth  as  great  as  that  of  the  ordina- 
ry polar  instrument.  The  theory  of  its  action  is  essentially 
the  same  as  the  other. 

147.  The  Rolling  Planimeter  is  the  most  accurate  instru- 
ment of  its  kind  yet  devised.  Its  compass  is  also  indefinitely 
increased,  since  it  may  be  rolled  bodily  over  the  sheet  for  any 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  1 53 


distance,  on  a right  line,  and  an  area  determined  within  certain 
limits  on  either  side.  It  is  therefore  especially  adapted  to  the 
measuring  of  cross-sections,  profiles,  or  any  long  and  narrow 
surface.  Fig.  38  shows  one  form  of  this  instrument  as  de- 
signed by  Herr  Corradi  of  Zurich.  It  is  a suspended  planim- 
eter,  inasmuch  as  the  wheel  rolls  on  a flat  disk  which  is  a 
part  of  the  instrument,  but  it  could  not  be  called  a polar  pla- 
nimeter,  the  theory  of  its  action  being  very  different  from  that 
instrument.  The  frame  B is  supported  by  the  shaft  carrying 


Fig.  38. 


the  two  rollers  To  this  frame  are  fitted  the  disk  A and  the 
axis  of  the  tracing-arm  F.  The  whole  apparatus  may  thus  move 
to  and  fro  indefinitely  in  a straight  line  on  the  two  rollers  while 
the  tracing-point  traverses  the  perimeter  of  the  area  to  be 
measured.  The  shaft  carries  a bevel-gear  wheel,  which 
moves  the  pinion  R^.  This  pinion  is  fixed  to  the  axis  of  the 
disk,  and  turns  with  it,  so  that  any  motion  of  the  rollers  7?, 
causes  the  disk  to  revolve  a proportional  amount,  and  the 
component  of  this  motion  at  right  angles  to  the  axis  of  the 
wheel  E is  recorded  on  that  wheel.  If  the  instrument  remains 


154 


SUR  VE  YING. 


stationary  on  the  paper  (the  rollers  R not  turning)  and  the 
tracing-point  moved  laterally,  it  will  cause  no  motion  of  the 
wheel,  since  its  axis  is  parallel  to  the  arm  F,  and  turns  about 
the  same  axis  with  F,  but  90°  from  it ; the  wheel  E,  therefore, 
moves  parallel  with  its  axis  and  does  not  turn. 

148.  Theory  of  the  Rolling  Planimeter. — This  will  be 
developed  by  a system  of  rectangular  coordinates,  the  path  of 
the  fulcrum  of  the  tracing-arm  being  taken  as  the  axis  of 


Fig.  39. 


abscissae.  The  path  of  the  tracing-point  will  be  considered 
as  made  up  of  two  motions,  one  parallel  to  the  axis  of  abscis- 
sae and  the  other  at  right  angles  to  it.  The  elementary  area 
considered  will  be  that  included  between  the  axis  of  abscissae 
and  two  ordinates  drawn  to  the  extremities  of  an  elementary 
portion  of  the  path.  But  since  this  element  of  the  perimeter 
is  supposed  to  be  made  up  of  two  right  lines,  one  perpendicu- 
lar to  the  axis  of  abscissae  and  the  other  parallel  to  it,  our 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  I55 


elementary  area  must  also  be  divided  in  a similar  manner. 
It  will  at  once  be  seen  that  one  part  of  this  area  is  zero,  since 
the  two  ordinates  bounding  it  form  one  and  the  same  line. 
This  is  the  part  generated  by  the  motion  at  right  angles  to 
the  axis  of  abscissae.  Now,  we  have  just  shown  in  the  previous 
article  that  the  wheel-record  for  this  part  of  the  path  is  also 
zero.  We  are  brought  therefore  to  this  important  conclusion  : 
that  all  components  of  motion  of  the  tracing-pomt  at  right  angles 
to  the  axis  of  abscisses  have  no  influence  upon  the  re  suit.  We 
will  therefore  only  discuss  a differential  motion  of  the  tracing- 
point  in  the  direction  of  the  axis  of  abscissae. 

In  Fig.  39,  which  is  a linear  sketch  of  the  instrument  shown 
in  Fig.  38,  with  the  corresponding  parts  similarly  lettered,  it 
is  to  be  shown  that  the  motion  of  the  wheel  E caused  by  the 
movement  of  the  tracing-point  over  the  path  dx  is  equal  to 
the  corresponding  area_y<^.r  multiplied  by  some  constant  which 
is  a function  of  the  dimensions  of  the  instrument. 

It  is  evident  that  a motion  of  the  tracing-point  in  the  di- 
rection of  the  axis  of  abscissae  can  only  be  obtained  by  moving 
the  entire  instrument  on  the  rollers  by  the  same  amount,  and 
therefore  when  the  point  moves  over  the  path  dx  the  circum- 
ferences of  the  rollers  have  moved  the  same  amount.  This 

causes  a movement  of  the  pitch  circle  of  of  dx  This 

motion  is  conveyed  to  the  disk  through  R^,  so  that  any  point 
on  this  disk,  as  a,  distant  ad  from  the  axis,  moves  through  a 
R ad 

distance  equal  to  dx  Let  aby  Fig.  39,  be  this  distance, 

Then  we  have 


ab  = dx^ 


ad 


(I) 


The  motion  of  that  portion  of  the  disk  on  which  the  roller 
rests,  equal  to  ab,  causes  the  circumference  of  the  wheel  E to 


156 


SUR  VE  YING. 


revolve  by  an  amount  equal  to  the  component  of  the  distance 
ab  perpendicular  to  the  axis  of  the  wheel.  Tliis  component 
part  of  the  disk’s  motion  is  bc^  and  this  is  the  measure  of  the 
wheel’s  motion.  It  therefore  remains  to  show  that  be  ■=^ ydx 
multiplied  by  an  instrumental  constant. 

Now,  be  — ab  sin  bae (2) 

But  bae  = a since  gae  and  bad  are  both  right  angles. 

Also,  bae  — supplement  of  dag=  a -|-  ft- 
Also,  from  the  triangle  dag,  we  have 


or 


sin  dag : sin  agd  ::  D : ad, 
D a 

sm  («  + /?)  = 


(3) 


Since  Fga  is  also  a right  angle,  we  have  the  angle  formed 

y 

by  Fg  and  the  axis  of  abscissae  equal  to  a,  whence  sin  a = 

We  may  now  write  : 

bc  = ab^,x.{a^^)  = ab--^-^  = ab^^.  . (4) 


Now,  substituting  the  value  of  ab  from  (i),  we  have 


eb  — ydx 


D . R, 
F^R,-R, 


is) 


Since  D,  R^,  F,  R,,  and  R^  are  all  constants  for  any  one 
instrument,  we  see  that  the  wheel-record  is  a function  of  the 
area  generated  by  the  tracing-point  and  the  instrumental  con- 
stants, which  was  to  be  shown.  It  now  follows  that  the  sum- 
mation of  all  these  elementary  areas  included  between  the 
path  of  the  tracing-point,  the  limiting  ordinates,  and  the  axis 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  157 


of  abscissae,  is  represented  by  the  total  wheel-movement , or  the 
difference  between  its  initial  and  final  readings.  If,  therefore, 
the  area  to  be  measured  is  of  this  character,  being  bounded  by 
one  right  line  and  limiting  ordinates,  it  would  not  be  necessary 
to  move  the  point  over  the  entire  perimeter,  but  only  along 
the  irregular  boundary,  provided  the  instrument  could  be  ad- 
justed with  the  point  ^ exactly  over  the  base  of  the  figure,  and 
with  the  axis  B at  right  angles  to  it,  so  that  in  rolling  the  in- 
strument along,  the  point  g would  remain  over  the  base-line. 
In  other  words,  the  axis  of  abscissa  of  the  instrument  would 
have  to  coincide  exactly  with  this  base-line.  Then  for  motion 
of  the  tracing-point  over  this  line,  as  well  as  for  its  motion  over 
the  end-ordinates,  the  wheel  would  not  revolve,  neither  would 
there  be  any  area  generated  between  these  lines  and  the  axis. 
In  general  this  cannot  be  done,  and  it  is  only  mentioned  here 
in  order  to  more  clearly  illustrate  the  working  of  the  instru- 
ment. 

As  in  the  case  of  the  polar  instrument,  the  proper  length  of 
arm  F,  to  be  used  with  the  rolling-planimeter  to  give  results 
in  any  desired  unit,  depends  on  the  other  instrumental  con- 
stants. These  being  known,  the  value  of  A may  be  computed 
in  the  same  manner  as  with  the  polar  planimeter. 

149.  To  test  the  Accuracy  of  the  Planimeter,  there  is 
usually  provided  a brass  scale  perforated  with  small  holes.  A 
needle-point  is  inserted  in  one  of  these  and  made  fast  to  the 
paper  or  board,  while  the  tracing-point  rests  in  another.  The 
latter  may  now  be  moved  over  a fixed  path  with  accuracy. 
Make  a certain  number  of  even  revolutions  forward,  or  in  the 
direction  of  the  hands  of  a watch,  noting  the  initial  and  final 
readings.  Reverse  the  motion  the  same  number  of  revolutions, 
and  see  if  it  comes  back  to  the  first  reading.  If  not,  the  dis- 
crepancy is  the  combined  instrumental  error  from  two  meas- 
urements due  to  slip,  lost  motion,  unevenness  of  paper,  etc. 

If  this  test  be  repeated  with  the  areas  on  opposite  sides  of 


158 


SURVEYING. 


the  zero-circle  in  the  case  of  the  polar-planimeter,  or  on  oppo- 
site sides  of  the  axis  of  abscissae  in  case  of  the  rolling-planimc- 
ter,  with  the  same  score  in  both  cases,  it  proves  that  tlie  pivot- 
points  a,  b,  k,  and  the  tracing-point  d (Fig.  35),  arc  in  the  same 
straight  line,  in  case  of  the  polar  instrument,  and  that  the  cor- 
responding points  in  the  suspended  and  rolling  planimctcrs 
form  parallel  lines;  in  other  words,  that  the  axis  of  the  meas- 
uring-wheel is  parallel  to  the  tracing-arm.  If  the  results  differ 
when  the  areas  lie  on  opposite  sides  of  the  axis  or  zero-circle, 
these  lines  are  not  parallel  and  must  be  adjusted  to  a parallel 
position. 

150.  Use  of  the  Planimeter. — The  paper  upon  which  the 
diap:ram  is  drawn  should  be  stretched  smooth  on  a level  sur- 
face.  It  should  be  large  enough  to  allow  the  rolling-wheel  to 
remain  on  the  sheet. 

The  instrument  should  be  so  adjusted  and  oiled  that  the 
parts  move  with  the  utmost  freedom  but  without  any  lost  mo- 
tion. This  requires  that  all  the  pivot-joints  shall  be  adjustable 
to  take  up  the  wear.  The  rim  of  the  measuring-wheel  must  be 
kept  bright  and  free  from  rust.  The  instrument  must  be  han- 
dled with  the  greatest  care.  Having  set  the  length  of  the 
tracing-arm  for  the  given  scale  and  unit,  it  is  well  to  test  it 
upon  an  area  of  known  dimensions  before  using.  If  it  be  found 

to  give  a result  in  error  by  ^ of  the  total  area,  the  length  of  the 

tracing-arm  must  be  changed  by  an  amount  equal  to  this  same 
ratio  of  its  former  length.  If  the  record  made  on  the  wheel 
was  too  small  then  the  length  of  the  tracing-arm  must  be  di- 
minished, and  vice  versa.  If  the  paper  has  shrunk  or  stretched, 
find  the  proportional  change,  and  change  the  length  of  the 
tracing-arm  from  its  true  length  as  Just  founds  by  this  same 
ratio,  making  the  arm  longer  for  stretch  and  shorter  for  shrink- 
age. Or  the  true  length  of  arm  may  be  used,  and  the  results 
corrected  for  change  in  paper. 


ADTUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  1 59 


To  measure  an  area,  first  determine  whether  the  fixed  point, 
or  pole,  shall  be  inside  or  outside  the  figure.  It  is  preferable 
to  have  it  outside  when  practicable,  since  then  the  area  is  ob- 
tained without  correction.  If,  however,  the  diagram  is  too 
large  for  this  (in  case  of  the  polar  planimeter)  the  pole  may  be 
set  inside.  In  either  case  inspection,  and  perhaps  trial,  is  nec- 
essary to  fix  upon  the  most  favorable  position  of  the  pole,  so 
that  the  tracing-point  may  most  readily  reach  all  parts  of  the 
perimeter.  If  the  area  is  too  large  for  a single  measurement, 
divide  it  by  right  lines  and  measure  the  parts  separately. 
Having  fixed  the  pole,  set  the  tracing-point  on  a well-defined 
portion  of  the  perimeter,  and  read  and  record  the  score  on 
the  rolling-wheel  and  disk.  This  is  generally  read  to  four 
places.  Move  the  tracing-point  carefully  and  slowly  over  the 
outline  of  the  figure,  in  the  direction  of  the  hands  of  a watch, 
around  to  the  initial  point.  Read  the  score  again.  If  the 
pole  is  outside  the  figure,  this  result  is  always  positive  when 
the  motion  has  been  in  the  direction  here  indicated.  If  the 
pole  is  inside  the  figure,  the  result  will  be  negative  when  the 
area  is  less  than  that  of  the  zero-circle,  positive  if  greater. 
With  the  pole  inside  the  figure,  however,  the  area  of  the  zero- 
circle  must  always  be  added  to  the  result  as  given  by  the  score, 
and  when  this  is  done  the  sum  is  always  positive,  the  motion 
being  in  the  direction  indicated.  The  area  of  this  zero-circle 
is  found  in  art.  144,  to  be  tt  + 2nl).  The  value 

of  /,  which  is  the  length  of  the  tracing-arm,  is  known.  The 
values  of  in  and  n should  be  furnished  by  the  maker.  If  these 
are  unknown,  the  area  of  the  zero-circle  can  be  found  for  any 
length  of  arm  /,  by  measuring  a given  area  with  pole  outside 
and  inside,  the  difference  in  the  two  scores  being  the  area  of 
this  circle.  By  doing  this  with  two  very  different  values  of  / 
we  may  obtain  two  equations  with  two  unknown  quantities, 
m and  n,  from  which  the  absolute  values  of  these  quantities 
may  be  found.  Thus  we  would  have: 


i6o 


SURVEYING. 


A = 71  {m^  +r  + 2nl) ; 
^'  = ;r  (;«’  + P + 2;//')  ; 


whence 


7t 


wherein  /,  A,  and  A'  are  known.  The  values  of  in  and  n are 
then  readily  found. 

In  using  the  rolling-planimeter,  it  is  advisable  to  take  the 
initial  point  in  the  perimeter  on  the  axis  of  abscissae,  as  in  this 
position  any  small  motion  of  the  tracing-point  has  no  effect 
on  the  wheel,  and  so  there  is  no  error  due  to  the  initial  and 
final  positions  not  being  exactly  identical. 

The  planimeter  may  be  used  to  great  advantage  in  the 
solution  of  many  problems  not  pertaining  to  surveying.  In 
all  cases  where  the  result  can  be  represented  as  a function 
of  the  product  of  two  variables  and  one  or  more  constants,  the 
corresponding  values  of  the  variables  may  be  plotted  on  cross- 
section  paper  by  rectangular  coordinates,  thus  forming  with 
the  axis  and  end-ordinates  an  area  which  can  be  evaluated  for 
any  scale  and  for  any  value  of  the  constant-functions  by  setting 
off  the  proper  length  of  tracing-arm.  Thus,  from  a steam- 
indicator  card  the  horse-power  of  the  engine  may  be  read  off, 
and  from  a properly  constructed  profile  the  amount  of  earth- 
work in  cubic  yards  in  a railway  cut  or  fill.  Some  of  these 
special  applications  are  further  explained  in  Part  II.  of  this 
work. 

151.  Accuracy  of  Planimeter-measurements. — Professor 
Lorber,  of  Loeben,  Austria,  has  thoroughly  investigated  the 
relative  accuracy  of  different  kinds  of  planimeters,  and  the  re- 
sults of  his  investigations  are  given  in  the  following  table.  It 


ADJUSTMENT,  USE,  AND  CAEE  OF  INSTRUMENTS.  l6l 


will  be  seen  that  the  relative  error  is  less  as  the  area  measured 
is  larger.  The  absolute  error  is  nearly  constant  for  all  areas,  in 
the  polar  planimeter.  The  remarkable  accuracy  of  the  rolling- 
planimeter  is  such  as  to  cause  it  to  be  ranked  as  an  instrument 
of  precision. 


TABLE  OF  RELATIVE  ERRORS  IN  PLANIMETER-MEASUREMENT3. 


Area  in — 

The  error  in  one  passage  of  the  tracer  amounts  on  an 
average  to  the  following  fraction  of  the  area  meas- 
ured by — 

The  ordinary  po- 
lar planimeter- 
Unit  of  vernier: 
10  sq.  mm. 

= .015  sq.  in. 

Suspended  plani- 
meter-Unit  ol 
vernier: 

I sq.  mm.  = 
.001  sq.  in. 

Rolling  planime- 
ter-Unit  of  ver- 
nier: 

I sq.  mm.  = 

.001  sq.  in. 

Square  cm. 

Square  inches. 

10 

1.55 

TZ 

20 

3.10 

TTTT 

zIwS: 

50 

7.75 

TZZ 

zluz 

W&T 

100 

15.50 

zrkz 

itVt 

TcVlF 

200 

31.00 

±27  f 

ttVj 

ITTS 

300 

46.50 

.... 

Wr? 

TGZZIS 

THE  PANTOGRAPH. 

152.  The  Pantograph  is  a kind  of  parallel  link-motion 
apparatus  whereby,  with  one  point  fixed,  two  other  points  are 
made  to  move  in  a plane  on  parallel  lines  in  any  direction. 
The  device  is  used  for  copying  drawings,  or  other  diagrams  to 
the  same,  a larger,  or  a smaller  scale.  The  theory  of  the  instru- 
ment rests  on  the  following: 

Proposition  : If  the  sides  of  a parallelogram,  jointed  at 
the  corners  A,  B,  C,  and  D,  and  indefinitely  extended,  be  cut  by 
a right  line  in  four  points,  as  E,  F,  G,  and  H,  then  these  latter 
pomts  will  lie  in  a straight  line  for  all  values  of  each  of  the 
parallelogram  angles  from  zero  to  180°,  and  the  ratio  of  the  dis- 
tances EF,  EG,  and  GH  will  remain  unchanged. 

II 


SURVEYING. 


162 


In  Fig.  40,  let  A,  B,  C,  Z>be  the  parallelogram,  whose  sides 
(extended)  are  cut  by  a right  line  in  F,  G,  E,  and  //.  It  is 
evident  that  one  point  in  the  figure  may  remain  fixed  while 


the  angles  of  the  parallelogram  change.  Let  this  point  be  G, 
Since  GC  and  GH,  radiating  from  G,  cut  the  parallel  lines 
DE  and  CH^  we  have 

GD  : DE  ::  GC  \ CH. 

Also,  for  similar  reasons, 

% 

ED  : DG  ::  EA  : AF. 

Now  since  the  sides  of  the  parallelogram,  as  well  as  all  the 
intercepts,  AF,  GD,  DE,  and  CH,  remain  constant  as  the  angles 
of  the  figure  change,  when  the  figure  has  taken  the  position 
shown  by  the  dotted  lines,  we  still  have 

GU  : UE  wGC  \ CH'-, 

also, 

E’U  : D'G  ::  EA' : A’F. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  163 


From  the  first  of  these  proportions  we  know  that  G,  E\ 
and  H'  are  in  the  same  straight  line,  and  the  same  for  G,  E' , 
and  F ; therefore,  they  are  all  four  in  the  same  straight  line. 

To  show  that  they  are  the  same  relative  distance  apart  as 
before  we  have, 

FG\GE\  EH  ::  BC  \ DE  \ CH^DE\ 


also, 


FG  : GE  : EH  ::  EC  : UE  : CH-UE. 


But 

BC  = EC,  DE  = UE,  and  CH^DE^  CH  - UE\ 
therefore  we  may  write, 

FG\GE\  EH  ::  FG  : GE  : EH, 

Q.  E.  D. 

It  is  evident  that  two  of  the  points  E,  F,  G,  and  -^T  may 
become  one  by  the  transversal  passing  through  the  point  of 
intersection  of  two  of  the  sides  of  the  parallelogram.  The 
above  proposition  would  then  hold  for  the  three  remaining 
points. 

In  the  Pantograph  only  three  of  the  four  points  E,  F, 
G,  and  H (Fig.  40)  are  used.  One  of  these  may  therefore 
be  taken  at  the  intersection  of  two  sides  of  the  parallelogram, 
but  it  is  not  necessarily  so  taken.  These  three  points  are:  the 
fixed  point,  the  tracing-point,  and  the  copying-point. 

In  Fig.  41,  i^is  the  fixed  point,  held  by  the  weight  P;  B is 
the  tracing-point,  and  D is  the  copying-point,  or  vice  versa  as 
to  B and  D.  The  parallelogram  is  E,  G,  B,  H.  The  points 


164 


SURVEYING. 


/%  and  D must  lie  in  a straight  line,  B being  at  the  inter- 
section of  two  of  the  sides  of  the  parallelogram.  The  points 
Ay  Ey  and  C are  supported  on  rollers.  In  Fig.  42,  the  fixed- 


Fig.  41. 


point  is  the  point  of  intersection  of  two  of  the  sides  of  the 
parallelogram.  The  upper  left-hand  member  of  the  frame  is 
not  essential  to  its  construction,  serving  simply  to  stiffen  the 


copying-arm,  the  fourth  side  of  the  parallelogram  being  the 
side  holding  the  tracing-point. 

In  Fig.  43,  neither  of  the  three  points  is  at  the  intersection 
of  two  of  the  sides  of  the  parallelogram,  and  hence  there  is  a 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  165 


fourth  point  unused,  having  the  same  properties  as  the  fixed, 
tracing,  and  copying  points,  it  being  at  the  intersection  of  the 
line  joining  these  three  points  with  the  fourth  side  of  the  par- 
allelogram. 

From  the  theoretical  discussion,  and  from  the  figures  shown, 
it  becomes  evident  that  there  may  be  an  indefinite  variety  of 


only  essential  conditions  are  that  the  fixed,  the  tracing,  and  the 
copying  points  shall  lie  in  a straight  line  on  at  least  three  sides 
of  a jointed  parallelogram,  either  point  serving  any  one  of  the 
three  purposes. 

153.  Use  of  the  Pantograph. — The  use  of  the  instrument 
is  easily  acquired.  Since  both  the  tracing  and  copying  points 
should  touch  the  paper  at  all  times,  such  a combination  as  that 
shown  in  Fig.  41  is  preferable  to  those  shown  in  Figs.  42  and 
43,  since  in  these  latter  the  tracing  point  is  surrounded  by  sup- 
ported points,  and  so  would  not  touch  the  paper  at  all  times 
unless  the  paper  rested  on  a true  plane.  In  most  instruments 
where  the  scale  is  adjustable,  the  two  corresponding  changes 
in  position  of  tracing  and  copying  points  for  different  scales 
are  indicated.  To  test  these  marks,  see  that  the  adjustable 
points  are  in  a straight  line  with  the  fixed  point,  and  to  test  the 

FD 

scale  see  that  the  ratio  (Fig.  41)  is  that  of  the  reduction 


desired.  Thus,  if  the  diagram  is  to  be  enlarged  to  twice  the 
original  size,  make  FD  = 2FB ; 


or  make 


DE_F^ 

DG  ~ ~BG 


= scale  of  enlargement. 


If  the  drawing  is  to  be  reduced  in  size,  make  B the  copying- 
point  and  D the  tracing-point. 

If  the  drawing  is  to  be  copied  to  the  same  scale,  make  BF 
— BD  and  make  B the  fixed  point.  The  figure  is  then  copied 
to  same  scale,  but  in  an  inverted  position. 

In  the  best  instruments  the  arms  are  made  of  brass,  but 
very  good  work  may  be  done  with  wooden  arms. 


PROTRACTORS. 

154.  A Protractor  is  a graduated  circle  or  arc,  with  its  cen- 
tre fixed,  to  be  used  in  plotting  angles.  They  are  of  various 
designs  and  materials. 

Semicircular  Protractors,  such  as  shown  in  Fig.  44,  are 

usually  made  of  horn,  brass,  or 
german-silver.  They  are  grad- 
uated to  degrees  or  half-degrees, 
and  the  angle  is  laid  off  by 
holding  the  centre  at  the  vertex 
of  the  angle,  with  the  plain 
edge,  or  the  o and  180  degree 
line  on  the  given  line  from  which 

the  angle  is  to  be  laid  off. 

In  the  full  circle  protractor,  shown  in  Fig.  45,  there  is  a 
movable  arm  with  a vernier  reading  to  from  i to  3 minutes. 
The  horn  centre  is  set  over  the  given  point,  the  protractor 
oriented  with  the  zero  of  the  circle  on  the  given  line,  and  the 
arm  set  to  the  given  reading  when  the  other  line  may  be 
drawn. 


ADJUSTMENT,  USE,  AND  CATE  OF  INSTRUMENTS.  1 6/ 


The  three-arm  protractor,  Fig.  46,  has  one  fixed  and  two 
movable  arms  by  which  two  angles  may  be  set  off  simulta- 
neously. It  is  used  in  plotting  observations  by  sextant  of  two 


Fig.  45. 


angles  to  three  known  points  for  the  location  of  the  point  of 
observation.  This  is  known  as  the  three-point  problem  and 
is  discussed  in  Chap.  X. 


Fig.  46. 


Paper  protractors  are  usually  full  circled,  from  8 to  14 
inches  in  diameter,  graduated  to  half  or  quarter  degrees. 
They  are  printed  from  engraved  plates  on  drawing-  or  tracing- 


5-6^/’  VE  Y I NG. 


1 68 


paper  or  bristol-board,  and  are  very  convenient  for  plotting 
topographical  surveys.  The  map  is  drawn  directly  on  the 
protractor  sheet,  the  bearing  of  any  line  being  taken  at  once 
from  the  graduated  circle  printed  on  the  paper.  These  “ pro- 
tractor sheets”  can  now  be  obtained  of  all  large  dealers. 

The  coordinate  protractor"^  is  a quadrant,  or  square,  with 


angular  graduations  on  its  circumference,  or  sides,  and  divided 
over  its  face  by  horizontal  and  vertical  lines,  like  cross-section 
paper.  A movable  arm  can  be  set  by  means  of  a vernier  to 
read  minutes  of  arc,  this  arm  being  also  graduated  to  read 
distances  from  the  centre  outward.  Having  set  this  arm  to 
read  the  proper  angle,  the  latitude  is  at  once  read  off  on  the 


* Called  a Trigonometer  by  Keuffel  & Esser,  the  makers. 


ADJUSTMENT,  USE,  AND  CARE  OF  INSTRUMENTS.  1 69 


vertical  scale  and  the  departure  on  the  horizontal  scale  for  the 
given  distance  as  taken  on  the  graduated  arm.  A quadrant 
protractor  giving  latitudes  and  departures  for  all  distances 
under  2500  feet  to  the  nearest  foot,  or  under  250  feet  to  the 
nearest  tenth  of  a foot,  has  been  used.  The  radius  of  the  cir- 
cle is  i8|  inches.  Both  the  protractor  and  the  arm  are  on 
heavy  bristol-board,  so  that  any  change  due  to  moisture  will 
affect  both  alike  and  so  eliminate  errors  due  to  this  cause. 
The  instrument  was  designed  to  facilitate  the  plotting  of  the 
U.  S.  survey  of  the  Missouri  River.*  It  has  proved  very 
efficient  and  satisfactory.  A similar  one  on  metal,  shown  in 
Fig.  47,  is  now  manufactured,  and  serves  the  same  purpose. 

PARALLEL  RULERS. 

155*  The  Parallel  Ruler  of  greatest  efficiency  in  plotting 
is  that  on  rollers,  as  shown  in  Fig.  48.  The  rollers  are  made 
of  exactly  the  same  circumfer- 
ence, both  being  rigidly  attached 
to  the  same  axis.  It  should  be 
made  of  metal  so  as  to  add  to  its 
weight  and  prevent  slipping.  It  is  of  especial  value  in  connec- 
tion with  the  paper  protractors,  for  the  parallel  ruler  is  set  on 
any  given  bearing  and  then  this  transferred  to  any  part  of  the 
sheet  by  simply  running  the  ruler  to  place.  Two  triangles 
may  be  made  to  serve  the  same  purpose,  but  they  are  not  so 
rapid  or  convenient,  and  are  more  liable  to  slip.  The  parallel 
ruler  is  also  very  valuable  in  the  solution  of  problems  in 
graphical  statics. 

SCALES. 

156.  Scales  are  used  for  obtaining  the  distance  on  the 
drawing  or  plot  which  corresponds  to  given  distances  on  the 


Fig.  48. 


For  sale  by  A.  S.  Aloe  & Co.,  St.  Louis,  Mo. 


170 


SURVEYmC. 


object  or  in  the  field.  There  is  such  a variety  of  units  for 
both  field  and  office  work,  and  a corresponding  variety  of 
scales,  that  the  choice  of  the  particular  kind  of  scale  for  any 
given  kind  of  work  needs  to  be  carefully  made.  Architects 
usually  make  the  scale  of  their  drawings  so  many  feet  to  the 
inch,  giving  rise  to  a duodecimal  scale,  or  some  multiple  of 


A surveyor  who  uses  a Gunter’s  chain  66  feet  in  length  plots 
his  work  to  so  many  chains  to  the  inch,  making  a scale  of 
some  multiple  of  engineer  usually  uses  a lOO-foot 

chain  and  a level  rod  divided  to  decimal  parts  of  a foot ; so  he 
finds  it  convenient  to  use  a decimal  scale  for  his  maps  and 
drawings,  reduced  to  the  inch-unit  however.  Here  the  field- 
unit  is  feet  and  the  office-unit  is  inches,  both  divided  deci- 
mally. This  gives  rise  to  a sort  of  decimal-duodecimal  system, 
the  scale  being  some  multiple  of  Various  combinations 

of  all  these  systems  are  found. 

Figure  49  shows  one  form  of  an  ivory  scale  of  equal  parts 
for  the  general  draughtsman.  The  lower  half  of  the  scale  is 
designed  to  give  distances  on  the  drawing  for  4,  40,  or  400 
units  to  the  inch  when  the  left  oblique  lines  and  bottom 
figures  are  used,  and  for  2,  20,  or  200  units  to  the  inch  when 
the  right  oblique  lines  and  top  figures  are  used.  Thus,  if  we 
are  plotting  to  a scale  of  400  feet  to  the  inch,  and  the  dis- 
tance is  564  feet,  set  one  point  of  the  dividers  on  the  vertical 
line  marked  5,  and  on  the  fourth  horizontal  line  from  the  bot- 
tom. Set  the  other  leg  at  the  intersection  of  the  sixth  inclined 


ADJUSl'MENT,  USE,  AND  CARE  OF  INSTRUMENTS.  IJl 


line  with  this  same  horizontal  line,  and  the  space  subtended  by 
the  points  of  the  dividers  is  564  feet  to  a scale  of 

Figure  50  is  a cut  of  an  engineer’s  triangular  boxwood 
scale,  12  inches  long,  being  divided  into  decimal  inches. 
There  are  six  scales  on  this  rule,  a tenth  of  an  inch  being  sub- 
divided into  I,  2,  3,  4,  5,  and  6 parts,  making  the  smallest 


Fig,  50. 


graduations  of  an  inch  respectively.  This 

is  called  an  engineer’s  or  decimal-inch  scale  The  architect’s 
triangular  scale  is  divided  to  give  J,  f,  f,  I,  ij,  2,  3,  and 
4 inches  to  the  foot.  Such  a scale  is  of  less  service  to  the 
civil  engineer. 


BOOK  II. 

SURVEYING  METHODS. 


CHAPTER  VII. 

LAND-SURVEYING. 

157.  Land-surveying  includes  laying-out,  subdividing, 
and  finding  the  area  of,  given  tracts  of  land.  In  all  cases  the 
boundary-  and  dividing-lines  are  the  traces  of  vertical  planes 
on  the  surface  of  the  ground,  and  the  area  is  the  area  of  the 
horizontal  plane  included  between  the  bounding  vertical 
planes.  In  other  words,  the  area  sought  is  the  area  of  the 
horizontal  projection  of  the  real  surface. 

158.  In  laying  out  Land  the  work  consists  in  running 
the  bounding-  and  dividing-lines  over  all  the  irregularities  of 
the  surface,  leaving  such  temporary  and  permanent  marks  as 
the  work  may  demand.  These  lines  to  lie  in  vertical  planes, 
and  their  bearings  and  horizontal  distances  to  be  found.  The 
bearing  of  a line  is  the  horizontal  angle  it  makes  with  a merid- 
ian plane  through  one  extremity,  and  its  length  is  the  length 
of  its  horizontal  projection.  This  reduces  the  plot  of  the  work 
to  what  it  would  be  if  the  ground  were  perfectly  level.  If  all 
the  straight  lines  of  a land-survey  lie  in  vertical  planes,  and  if 
their  bearings  and  horizontal  lengths  are  accurately  deter, 
mined,  then  as  a land  survey  it  is  theoretically  perfect,  what- 
ever the  purpose  of  the  survey  may  be. 


4 


ZAJVD  SUJ^FEYING. 


173 


THE  UNITED  STATES  SYSTEM  OF  LAYING  OUT  THE  PUBLIC 

LANDS. 

159*  The  Public  Lands  of  the  United  States  have 
included  all  of  that  portion  of  our  territory  north  of  the  Ohio 
River  and  west  of  the  Mississippi  River  not  owned  by  indi- 
viduals previous  to  the  dates  of  cession  to  the  United  States 
Government.  All  of  this  territory,  except  the  private  claims, 
has  been  subdivided,  or  laid  out,  in  rectangular  tracts  bounded 
by  north  and  south  and  east  and  west  lines,  each  tract  having 
a particular  designation,  such  that  it  is  impossible  for  the  pat- 
ents or  titles,  as  obtained  from  the  Government,  to  conflict. 
This  has  saved  millions  of  dollars  to  the  land-owners  in  these 
regions  by  preventing  the  litigations  that  are  common  in  the 
old  colonial  States,  and  is  one  of  the  greatest  boons  of  our 
national  Government.  The  system  was  probably  devised  by 
Gen.  Rufus  Putnam,*  an  American  officer  in  the  Revolution- 
ary War.  It  was  first  used  in  laying  out  the  eastern  portion 
of  the  State  of  Ohio,  in  1786-7,  then  called  the  Northwest 
Territory.  This  was  the  first  land  owned  and  sold  by  the 
national  Government.  The  details  of  the  system  have  been 
modified  from  time  to  time,  but  it  remains  substantially  un- 
changed. The  following  is  a synopsis  of  the  method  which  is 
given  in  detail  in  the  Instructions  to  Survey  or s-General,  issued 
by  the  Commissioner  of  the  General  Land  Office,  at  Washing- 
ton, D.  C.,  and  obtained  on  application. 

160.  The  Reference-lines  consist  of  Principal  Meridians 
and  Standard  Parallels.  The  principal  meridians  may  be  a 
hundred  miles  or  more  apart,  but  the  standard  parallels  are  24 
miles  apart  north  of  35°  north  latitude,  and  30  miles  apart 
south  of  that  line.  These  lines  should  be  run  with  great  care, 


* See  article  by  Col.  H.  C.  Moore  in  Journal  of  the  Association  of  En- 
gineering Societies,  vol.  ii.,  p.  282. 


174 


SURVEYING. 


using  the  solar  compass  or  solar  attachment.  The  magnetic 
needle  cannot  be  relied  on  for  this  work,  for  two  reasons  : there 
may  be  local  attraction  from  magnetic  deposits,  and  the  dec- 
lination changes  rapidly  (about  a minute  to  the  mile)  on  east 
and  west  lines.  The  transit  alone  might  be  used  to  run  out 
the  meridians,  as  this  consists  simply  of  extending  a line  in  a 
given  direction.  The  transit  could  not  boused  for  running  the 
parallels,  however,  for  these  are  ever  changing  their  direction, 
since  they  are  at  all  points  perpendicular  to  the  meridian  at 
that  point.  This  change  in  direction  is  due  to  the  convergence 
of  the  meridians.  The  solar  compass  is  the  only  surveying  in- 
strument that  can  be  used  for  running  a true  east  and  west 
line  an  indefinite  distance.  The  needle-compass  would  do  if 
there  were  no  local  attraction  and  if  the  true  declination  were 
known  and  allowed  for  at  all  points.  The  solar  compass  (or 
solar  attachment)  is  the  instrument  recommended  for  this 
work. 

In  running  these  reference-lines,  every  8o  chains  (every  mile) 
is  marked  by  a stone,  tree,  mound,  or  other  device,  and  is 
called  a “ section  corner.”  Every  sixth  mile  has  a different 
mark,  and  is  called  a “ township  corner.” 

i6i.  The  Division  into  Townships. — From  each  “town- 
ship corner”  on  any  standard  parallel  auxiliary  meridians  are 
run  north  to  the  next  standard  parallel.  Since  these  meridians 
converge  somewhat  towards  the  principal  meridian,  they  will 
not  be  quite  a mile  apart  when  they  reach  the  next  standard 
parallel.  But  the  full  six-mile  distances  have  been  marked  off 
on  this  parallel  from  the  principal  meridian,  and  it  is  from  these 
township  corners  that  the  next  auxiliary  meridians  will  start 
and  run  north  to  the  next  standard  parallel,  etc.  Thus  each 
standard  parallel  becomes  a “ correction-line”  for  the  merid- 
ians. The  territory  has  now  been  divided  into  “ranges” 
which  are  six  miles  wide  and  twenty-four  miles  long,  each  range 
being  numbered  east  and  west  from  the  principal  meridian. 


LAND  SURVEYING. 


175 


These  ranges  are  then  cut  by  east  and  west  lines  joining  the 
corresponding  township  corners  on  the  meridians,  thus  dividing 
the  territory  into  “ townships,'’  each  six  miles  square,  neglect- 
ing the  narrowing  effect  of  the  convergence  of  the  meridians. 
The  townships  are  numbered  north  and  south  of  a chosen 
parallel,  which  thus  becomes  the  “ Principal  Base-line.”  The 
fifth  township  north  of  this  base-line,  lying  in  the  third  range 
west  of  the  principal  meridian,  would  be  designated  as  “ town 
five  north,  range  three  west.”  Each  township  contains  thirty- 
six  square  miles,  or  23,040  acres. 

162.  The  Division  into  Sections. — The  township  is  di- 
vided into  thirty-six  sections,  each  one  mile  square  and  contain- 
ing 640  acres.  This  is  done  by  beginning  on  the  south  side  of 
each  township  and  running  meridian  lines  north  from  the  sec- 
tion corners”  already  set,  marking  every  mile  or  “ section 
corner,”  and  also  every  half-mile  or  “ quarter-section  corner.” 
When  the  fifth  section  corner  is  reached,  a straight  line  is  run 
to  the  corresponding  section  corner  on  the  next  township  line. 
This  will  cause  this  bearing  to  be  west  of  north  on  the  west,  and 
east  of  north  on  the  east,  of  the  principal  meridian.  When  this 
northern  township  boundary  is  a standard  or  correction-line, 
then  the  sectional  meridians  are  run  straight  out  to  it,  and  thus 
this  line  becomes  a correction-line  for  the  section-lines  as  well 
as  for  the  township-lines.  The  east  and  west  division-lines  are 
then  run,  connecting  the  corresponding  section  corners  on  the 
meridian  section  lines,  always  marking  the  middle,  or  quarter- 
section  points.  Evidently,  to  run  a straight  line  between  two 
points  not  visible  from  each  other,  it  is  necessary  first  to  run  a 
random  or  trial  line,  and  to  note  the  discrepancy  at  the  second 
point.  From  this  the  true  bearing  can  be  computed  and  the 
course  rerun,  or  the  points  on  the  first  course  can  be  set  over 
the  proper  distance.  The  sections  are  numbered  as  shown  in 
Figs.  51  and  52. 

When  account  is  taken  of  the  convergence  of  meridians,  the 


SUR  VE  YING. 


176 


sections  in  the  northern  tiers  of  each  township  will  not  be  quite 
one  mile  wide,  east  and  west  ; but  as  the  section  corners  are  set 
at  the  full  mile  distances  on  the  township-lines,  the  southern 
sections  in  the  next  town  north  begin  again  a full  mile  in  width. 
In  setting  the  section  and  quarter-section  corners  on  the  cast 
and  west  town  lines  the  full  distances  are  given  from  the  east 
towards  the  west  across  each  township,  leaving  the  deficiency 
on  the  last  quarter-section,  or  40-chain  distance,  until  the  next 
correction-line  is  reached,  when  the  town  meridians  are  again 
adjusted  to  the  full  six-mile  distances. 

163.  The  Convergence  of  the  Meridians  is,  in  angular 
amount,* 

c = m sin  ^ {L  L) ; 

where  in  — meridian  distance  in  degrees,  or  difference  of  longi- 
tude, and  L and  L are  the  latitudes  of  the  two  positions.  In 
other  words,  the  angular  convergence  of  the  meridians  is  the 
difference  in  longitude  into  the  sine  of  the  mean  latitude. 

The  convergence  in  chains  of  two  township-lines  six  miles 
apart,  from  one  correction-line  to  another  twenty-four  miles 
apart,  in  lat.  40°,  is 


C = 24  X 80  X sin  iT ; 

where  in  degrees,  = -g^  sin  40°,  since  one  degree  of  longitude 
in  lat.  40°  = 53  miles.  Thus  ^ = 4^.37  for  each  six-mile  dis- 
tance, east  or  west,  in  lat.  40°.  Whence  C = 2.42  chains, 
which  is  what  the  northern  tier  of  sections  in  the  north  range 
between  correction-lines  lacks  of  being  six  miles  east  and  west. 

In  a similar  manner,  we  may  find  that  the  north  sections  in 
a town  are  about  six  feet  narrower,  east  and  west,  than  the 
corresponding  southern  sections  in  the  same  town. 


* From  Eq.  (G),  p.  621,  when  cos  \Ah\s  taken  as  unity. 


LAND  SURVEYING. 


177 


Figures  51  and  52  show  the  resulting  dimensions  of  sections 
in  chiains  when  no  errors  are  made  in  the  field-work.  The 
north  and  south  distances  are  all  full  miles.  , 


Fig.  51. 


79.40 

80 

80 

80 

80 

80 

6 

5 

4 

3 

2 

I 

79.92 

79.92 

79-92 

79.92 

79-92 

79-92 

7 

8 

9 

10 

II 

12 

79-94 

79-94 

18 

17 

16 

15 

14 

13 

79-95 

79-95 

19 

20 

21 

22 

23 

24 

79-97 

79-97 

30 

29 

28 

27 

26 

25 

’ 79-98 

79-98 

31 

32 

34 

34 

35 

36 

80 

80 

80 

80 

80 

80 

CORRECTION-LINE. 


In  Fig.  51  it  will  be  observed  that  in  the  northern  tier  of 
sections  the  meridians  must  bear  westerly  somewhat  so  as  to 
meet  the  full-mile  distance,  laid  off  on  the  township-line. 

In  Fig.  52  they  continue  straight  north  to  the  town-line, 
which  is  in  this  case  a correction-line.  If  the  distances  on  this 
correction-line  be  summed  they  will  be  found  to  be  2.42  chains 
short  of  six  miles  as  above  computed. 

The  law  provides  that  all  excesses  or  deficiencies,  either 
12 


178 


SU/!  VE  YING. 


CORRF-CTION-I.INF.. 


78.08 

6 

78.10 

79.90 

5 

79,90 

4 

79.90 

3 

79.90 

2 

79.90 

79.92 

7 

78.12 

8 

9 

10 

II 

12 

79-94 

18 

78.13 

17 

16 

15 

14 

13 

79-95 

19 

78.14 

20 

21 

22 

23 

24 

79-97 

30 

78.16 

29 

28 

27 

26 

25 

79.98 

31 

32 

33 

34 

35 

36 

78.18 

80 

80 

80 

80 

80 

Fig.  52. 


from  erroneous  measurements  or  bearings  or  from  the  conver- 
gence of  meridians,  shall,  so  far  as  possible,  be  thrown  into  the 
northern  and  western  quarter-sections  of  the  township. 

164.  Corner  Monuments  have  been  established  on  all 
United  States  land  surveys  at  the  corners  of  townships,  sec- 
tions, and  quarter-sections,  except  at  the  quarter-section  corner 
at  the  Centre  of  each  section.  These  corners  have  consisted 
of  stones,  trees,  posts,  and  mounds  of  earth.  Witness-  or  bear- 
ing-trees have  always  been  blazed  and  lettered  for  the  given 
town,  range,  and  section,  one  tree  in  each  section  or  town 
meeting  at  that  corner,  whenever  such  trees  were  available. 
The  bearings  and  distances  to  such  trees,  and  a description  of 


LAND  SURVEYING. 


179 


the  same,  are  given  in  the  field-notes.  All  such  corners  and 
witness  points,  except  those  made  of  stone,  are  subject  to  de- 
struction and  decay,  and  when  these  are  lost  there  is  no  means 
of  relocating  the  boundary-lines.  They  were  designed  to  serve 
only  until  the  land  should  be  sold  off  to  individuals,  when  it 
was  expected  the  owner  would  replace  them  with  marks  of  a 
more  permanent  character.  This  has  seldom  been  done,  so 
that  in  many  instances  the  sectional  boundaries  can  now  only 
be  redetermined  by  personal  testimony,  line  fences  and  other 
circumstantial  evidence.* 

FINDING  THE  AREA  OR  SUPERFICIAL  CONTENTS  OF  LAND 
WHEN  THE  LIMITING  BOUNDARIES  ARE  GIVEN. 

165.  The  Area  of  a Piece  of  Land  is  the  area  of  the  level 
surface  included  within  the  vertical  planes  through  the  bound- 
ary-lines. This  area  is  found  in  acres,  roods,  and  perches,  or, 
better,  in  acres  only,  the  fractional  part  being  expressed 
decimally.  Evidently  the  finding  of  such  an  area  involves  two 
distinct  operations,  viz. : the  Field-work,  to  determine  the 
positions,  directions,  and  lengths  of  the  boundary-lines ; and 
the  Computation,  to  find  the  area  from  the  field-notes.  There 
are  several  methods  of  making  the  field  observations,  giving 
rise  to  corresponding  methods  of  computation.  Thus,  the 
area  maybe  divided  into  triangles,  and  the  lengths  of  the  sides, 
or  the  angles  and  one  side,  or  the  bases  and  altitudes  measured, 
and  the  several  partial  areas  computed.  Or  the  bearings  and 
distances  of  the  outside  boundary-lines  maybe  determined  and 
the  included  area  computed  directly.  This  is  the  common 
method  employed.  Again,  the  rectangular  coordinates  of  each 
of  the  corners  of  the  tract  may  be  found  in  any  manner  with 
reference  to  a chosen  point  which  may  or  may  not  be  a point 
in  the  boundary,  and  the  area  computed  from  these  coordi- 
nates. These  three  methods  will  be  described  in  detail. 


* See  Appendix  A. 


i8o 


SU/^VEVING. 


I.  Area  by  Triangular  Subdivision. 

i66.  By  the  Use  of  the  Chain  Alone. — In  Fig.  53  let 
ABCDEF  be  the  corner  bound- 
aries of  a tract  of  land,  the  sides 
being  straight  lines.  Measure 
all  the  sides  and  also  the  diag- 
onals AC,  AD,  AE,  and  FB. 

The  area  required  is  then  the 
sum  of  the  areas  of  the  four  tri- 
angles ABC,  A CD,  ADE,  and 
AEF.  These  partial  areas  are 
computed  by  the  formula 

Area  = Vs{s  — a){s  — b){s  — c), 

where  s is  the  half  sum  of  the  three  sides  a,  b,  c in  each  case. 

For  a Check,  plot  the  work  from  the  field-notes.  Thus,  take 
any  point  as  A and  draw  arcs  of  circles,  with  A as  the  com- 
mon centre,  with  the  radii  AB,  AC,  AD,  AE,  and  ^ A' taken  to 
the  scale  of  the  plot.  From  any  point  on  the  first  arc,  as  B, 
and  with  a radius  equal  to  BC  to  scale,  cut  the  next  arc,  whose 
radius  was  AC,  giving  the  point  C.  From  C find  D with  the 
measured  distance  CD,  etc.,  until  F is  reached.  Measure  FB 
on  the  plot,  and  if  this  is  equal  to  the  measured  length  of  this 
line,  taken  to  the  scale  of  the  drawing,  the  field-work  and  plot 
are  correct.  It  is  evident  the  point  A might  have  been  taken 
anywhere  inside  the  boundary-lines  without  changing  the 
method. 

167.  By  the  Use  of  the  Compass,  or  Transit,  and  Chain. 

— If  the  compass  had  been  set  up  at  A the  outer  boundaries 
could  have  been  dispensed  with,  except  the  lines  AB  and  AF. 
All  that  would  be  necessary  in  this  case  would  be  the  bear- 
ings and  distances  to  the  several  corners.  We  then  have  two 


LA//D  SURVEYING. 


i8i 


sides  and  the  included  angle  of  each  triangle  given  when  the 
area  of  each  triangle  is  found  by  the  formula: 

Area  = \ab  sin  C. 

In  this  case  there  is  no  check  on  the  chaining  or  bearings. 
I’he  taking-out  of  the  angles  from  the  given  bearings  could  be 
checked  by  summing  them.  This  sum  should  be  360°  when 
A is  inside  the  boundary-line,  and  360°  minus  the  exterior 
angle  FAB  when  A is  on  the  boundary.  If  the  boundary- 
lines be  measured  also,  then  the  area  of  each  triangle  can  be 
computed  by  both  the  above  methods  and  a check  obtained. 

168.  By  the  Use  of  the  Transit  and  Stadia.*— Set  up 
at  A,  or  at  any  interior  or  boundary  point  from  which  all  the 
corners  can  be  seen,  and  read  the  distances  to  these  corners 
and  the  horizontal  angles  subtended  by  them.  The  area  is 
then  computed  by  the  formula  given  in  the  previous  article. 
The  distances  may  be  checked  by  several  independent  read- 
ings, and  the  angles  by  closing  the  horizon  (sum  = 360°). 

The  above  methods  do  not  establish  boundary-lines,  which 
is  usually  an  essential  requirement  of  every  survey. 


II.  Area  from  Bearing  a7id  Length  of  the  Boundary-lines. 

169.  The  Common  Method  of  finding  land  areas  is  by 
means  of  a compass  and  chain.  The  bearings  and  lengths  of 
the  boundary-lines  are  found  by  following  around  the  tract  to 
the  point  of  beginning.  If  the  boundary-lines  are  unobstructed 
by  fences,  hedges,  or  the  like,  then  the  compass  is  set  at  the 
corners,  and  the  chaining  done  on  line.  If  these  lines  are  ob- 
structed, then  equal  rectangular  offsets  are  measured  and  the 


* The  stadia  methods  are  described  in  Chapter  VIII. 


i82 


SURVEYING. 


bearings  and  lengths  of  parallel  lines  are  determined.  In  this 
case  the  compass  positions  at  any  corner  for  the  two  courses 
meeting  at  that  corner  are  not  coincident,  neither  are  the  final 
point  of  one  course  and  the  initial  point  of  the  next  course, 
the  perpendicular  offsets  from  the  true  corner  overlapping  on 
angles  less  than  i8o°  and  separating  on  angles  over  i8o°. 

The  chaining  is  to  be  done  as  described  in  art.  4,  p.  8,  the 
66-foot  or  Gunter’s  chain  being  used.  Both  the  direct  and  the 
reverse  bearing  of  each  course  should  be  obtained  for  a check 
as  well  as  to  determine  the  existence  of  any  local  attraction. 
For  the  methods  of  handling  and  using  the  compass  see 
Chapter  II. 

170.  The  Field-notes  should  be  put  on  the  left-hand  page 
and  a sketch  of  the  line  and  objects  crossing  it  on  the  right- 
hand  page  of  the  note-book.  The  following  is  a convenient 
form  for  keeping  the  notes.  They  are  the  field-notes  of  the 
survey  which  is  plotted  on  p.  184.  It  will  be  seen  that  the 
“tree”  was  sighted  from  each  corner  of  the  survey  and  its 
bearing  recorded.  If  these  lines  were  plotted  on  the  map 
they  would  be  found  to  intersect  at  one  point.  If  the  plot 
had  not  closed,  then  these  bearings  would  have  been  plotted 
and  they  would  not  have  intersected  at  one  point,  the  first 
line  which  deviated  from  the  common  point  indicating  that 
the  preceding  course  had  been  erroneously  measured,  either  in 
bearing  or  distance,  or  else  plotted  wrongly.  In  general  such 
bearings,  taken  to  a common  point,  enable  us  to  locate  an 
error  either  in  the  field-notes  or  in  the  plot.  The  bearings  of 
all  division-fences  were  taken,  as  well  as  their  point  of  inter- 
section with  the  course,  so  that  these  interior  lines  could  be 
plotted  and  a map  of  the  f^rm  obtained.  The  “old  mill”  is 
located  by  bearings  taken  from  corners  .5  and  G.  The  reverse- 
bearings  are  given  in  parenthesis. 


LAATD  SURVEYING. 


183 


FIELD  NOTES— COMPASS  SURVEY.  Oct.  23,  1885. 


No.  of 
Course. 

Point. 

Bearing. 

Distance 
along  the 
Course. 

Remarks. 

S.  76°  50'  E.. . . 
West 

Ch. 

True  bearings  given. 
Variation  of  needle  5° 
50'  east. 

Henry  Flagg, 

Compassman. 
PeterLong,  } . 

John  Short, 

7 . 20 

Yard  “ 

9-75 

11.54 

13.90 

25.42 

I 

4 4 

Orchard  “ 

i< 

Corner  B 

South 

Wt.=  I 

(North) 

B.  T 

N.  54°  15'  E... 
N.  58°  E... 

North 

Courses  i and  2 are 
along  the  centres  of 
the  highway. 

Old  Mill 

2 

Fence 

12.50 

24.10 

34.68 

Corner  C 

S.  89°  55'  E.... 
(West) 

Wt.=  I 

B.  T 

N.  22°  20'  W . . 

Old  Mill 

N.  26°  45'  W.. . 

Fence 

N.  61°  45'  W... 

9.90 

10.70 

12.45 

24.00 

3 

Mill  Creek 

Fence 

N.  64°  W 

Corner  D 

N.  27°  40'  E. . . 
(S.  27“  45'  VV.) 

- 

1 

B.  T 

S.  85°  W 

N.  19°  10'  W. . 
(S.  19°  15'  E.).. 

4 

Corner  E 

7.40 

Wt.=  2 

B.  T 

S.  62°  30'  W... 
South 

C 

Fence 

15.80 

D 

Corner  F 

N.  86°  50'  W. . 

25.58 

Wt.=  2 

(S.  86°  45’  E.).. 

B.  T 

S.  40°  15'  E 

N.  bank  Mill  Creek. 

0.30 
0. 80 

6 

S.  “ 

Corner  G 

S.  47°  30'  W. . . 
(N.  47°  30'  E.). 

1.50 

Wt.=  5 

Fence 

S.  32°  E 

0.00 

Offset,  0.40 

0.00 

“ .60 

3.00 

“ .80 

6.00 

7 

“ .70 

9.00 

12.00 

• * •••••• 

* ‘ , 

“ .20 

13.60 

13.60 

Corner  H 

S.  77°  45'  W... 
(N.  77°  45'  E.). 

Wt.=  3 

8 

Corner  A 

S 89°  W 

3.53 

Wt.=  I 

(N.  89°  E.) 

SUR  VE  YING. 


184 


-Highway- 


LAND  SURVEYING. 


185 


COMPUTING  THE  AREA. 


Fig.  55. 


171.  The  Method  stated.  — In 
Fig.  55,*  let  ABODE  be  the  tract  whose 
area  is  desired.  Let  us  suppose  the 
bearings  and  lengths  of  the  several 
courses  have  been  observed.  Pass  a 
meridian  through  the  most  westerly 
corner,  which  in  this  case  is  the  corner 
A.  Let  fall  perpendiculars  upon  this 
meridian  from  the  several  corners,  and 
to  those  lines  drop  other  perpendicu- 
lars from  the  adjacent  corners,  as  shown 
in  the  figure.  Then  we  have: 


Area  ABODE  = bBODfb  - bBAEDfb 

= bBOe  + eODf  - (bBA  -f-  AEa  + aEDf).  (i) 


Hence  twice  the  area  ABODE  is 

2A  ^ (bB  + eO)Bc  + {eO  + fD)Dd 
— {bB)Ab  — (aE)Aa  — (^5*  -f-  fD)Eg.  ...  (2) 


We  will  now  proceed  to  show  that  these  distances  are  all 
readily  obtained  from  the  lengths  and  bearings  of  the  courses. 

172.  Latitudes,  Departures,  and  Meridian  Distances. — 
The  latitude  of  a course  is  the  length  of  the  orthographic  pro- 
jection of  that  course  on  the  meridian,  or  it  is  the  length  of  the 
course  into  the  cosine  of  its  bearing.  If  the  forward  bearing 
of  the  course  is  northward  its  latitude  is  called  its  7wrthing.  and 
is  reckoned  positively ; while  if  the  course  bears  southward  its 
latitude  is  called  its  southing,  and  is  reckoned  negatively. 


* The  lines  OD  and  OX  in  this  figure  are  used  in  art.  185. 


i86 


SURVEYING. 


The  departure  of  a course  is  the  length  of  its  orthographic 
projection  on  an  east  and  west  line,  or  it  is  the  length  of  the 
course  into  the  sine  of  its  bearing.  If  the  forward  bearing  of 
the  course  is  eastward  its  departure  is  called  its  easting,  and  is 
reckoned  positively  ; while  if  its  forward  bearing  is  westward 
its  departure  is  called  its  westing,  and  is  reckoned  negatively. 

The  meridian  distance  of  a point  is  its  perpendicular  dis- 
tance from  the  reference  meridian,  which  is  here  taken  through 
the  most  westerly  point  of  the  survey. 

The  meridian  distance  of  a course  is  the  meridian  distance 
of  the  middle  point  of  that  course  ; therefore 

The  double  meridian  distance  of  a eourse  is  equal  to  the  sum 
of  the  meridian  distances  to  the  extremities  of  that  course. 
The  D.  M.  D.’s  of  the  two  courses  adjacent  to  the  reference 
meridian  are  evidently  equal  to  their  respective  departures. 
The  D.  M.  D.  of  any  other  course  is  equal  to  the  D.  M.  D.  of 
the  preceding  course  plus  the  departure  of  that  course  plus 
the  departure  of  the  course  itself,  easterly  departures  being 
counted  positively  and  westerly  departures  negatively.  This 
is  evident  from  Fig.  55. 

Thus  in  Fig.  55  Dd  is  the  latitude  and  dC  is  the  departure 
of  the  course  DC.  If  the  survey  was  made  with  the  tract  on 
the  left  hand,  then  the  latitude  of  this  course  is  positive  and 
the  departure  negative ; while  the  reverse  holds  true  if  the 
survey  was  made  with  the  tract  on  the  right  hand.  In  this 
discussion  it  will  be  assumed  that  the  survey  is  made  by  going 
around  to  the  left,  or  by  keeping  the  tract  on  the  left  hand, 
although  this  is  not  essential.  The  D.  M.  D.  of  this  course 
CD  is  fD  -f  eC',  or  it  is  the  D.  M.  D.  of  BC  f-  cC  — dC). 

In  equation  (2),  art.  171,  the  quantities  enclosed  in  brack- 
ets are  the  double  meridian  distances  of  the  several  courses, 
all  of  which  are  positive,  while  the  distances  into  which  these 
are  multiplied  are  the  latitudes  of  the  corresponding  courses. 
If  we  go  around  towards  the  left  the  latitudes  of  the  courses 


LAND  SURVEYING. 


187 


AB^  DE,  and  EA  are  negative,  and  therefore  the  correspond- 
ing products  are  negative,  while  the  latitudes  of  the  courses 
BC  and  CD  being  positive,  their  products  are  positive. 

We  may  therefore  say  that  twice  the  area  of  the  figure  is 
equal  to  the  algebraic  sum  of  the  products  of  the  double  ineridia^i 
distances  of  the  several  courses  into  the  corresponding  latitudes, 
north  latitudes  being  reckoned  positively  and  south  latitudes 
negatively,  and  the  tract  being  kept  on  the  left  in  making  the 
survey.  If  the  tract  be  kept  on  the  right  in  the  survey,  then 
the  numerical  value  of  the  result  is  the  same,  but  it  comes  out 
with  a negative  sign. 

173.  Computing  the  Latitudes  and  Departures  of  the 
Courses. — Since  the  departure  of  a course  is  its  length  into 
the  sine,  and  its  latitude  its  length  into  the  cosine,  of  its  bear- 
ing, these  may  be  computed  at  once  from  a table  of  natural  or 
logarithmic  sines  and  cosines.  When  bearings  were  (formerly) 
read  only  to  the  nearest  15  minutes  of  arc,  tables  were  used 
giving  the  latitude  and  departure  for  all  bearings  expressed  in 
degrees  and  quarters  for  all  distances  from  i to  100.  Such 
tables  are  called  traverse  tables.  It  is  customary  now,  how- 
ever, to  read  even  the  needle-compass  closer  than  the  nearest 
15  minutes;  and  if  forward  and  back  readings  are  taken  on  all 
courses,  and  the  mean  used,  these  means  will  seldom  be  given 
in  even  quarters  of  a degree.  If  the  transit  or  solar  compass  is 
used,  the  bearing  is  read  to  the  nearest  minute.  The  old  style 
of  traverse  table  is  therefore  of  little  use  in  modern  survey- 
ing. The  ordinary  five-  or  six-place  logarithmic  tables  of 
sines  and  cosines  are  computed  for  each  minute  of  arc,  and 
these  may  be  used,  but  they  are  unnecessarily  accurate  for  or- 
dinary land-surveying.  For  this  purpose  a four-place  table  is 
sufficient.  If  the  average  error  of  the  field-work  is  as  much  as 
I in  1000  (and  it  is  usually  more  than  this),  then  an  accuracy 
of  I in  5000  in  the  reduction  is  evidently  all-sufficient,  and  this 
is  about  the  average  maximum  error  in  a four-place  table;  that 


r88 


SURVEYING. 


is,  the  average  of  the  maximum  errors  that  can  be  made  in  the 
different  parts  of  the  table. 

Table  III.  is  a four-place  table  of  logarithms  of  numbers 
from  I to  10,000,  and  Table  IV.  is  a similar  table  of  logarithms 
of  sines  and  cosines,  from  o to  360  degrees.  If  a transit  is 
used  in  making  the  survey,  and  if  it  is  graduated  continu- 
ously from  o to  360  degrees,  then  the  azimuths  of  the  several 
sides  are  found,  all  referred  to  the  true  meridian  or  to  the  first 
side.  If  it  is  desired  now  to  take  out  the  latitudes  and  de- 
partures, the  same  as  for  a compass-survey,  where  the  bearings 


N 

o 

190 


of  the  sides  are  given  directly  referred  to  the  north  and  south 
points,  it  may  be  done  by  Table  IV. 

Since  the  log  sine  changes  very  fast  near  zero  and  the  log 
cosine  very  fast  near  90°,  the  table  is  made  out  for  every  min- 
ute for  the  first  three  degrees  from  these  points ; for  the  rest 
of  the  quadrant  it  gives  values  10  minutes  apart,  but  with  a 
tabular  difference  for  each  minute.  It  is  very  desirable  to 
make  the  table  cover  as  few  pages  as  possible  for  convenience 
and  rapidity  in  computation.  In  this  table  the  zero-point  is 


LAND  SURVEYING. 


189 


south  and  angles  increase  in  the  direction  SWNE,  so  that  in 
the  first  quadrant  both  latitudes  and  departures  are  negative. 
In  the  second  quadrant  latitude  is  positive  and  departure  nega- 
tive, in  the  third  both  are  positive,  and  in  the  fourth  latitude 
is  negative  and  departure  positive.  These  relations  are  shown 
in  Fig.  56.  For  any  angle,  falling  in  any  quadrant,  if  reckoned 
from  the  south  point  in  the  direction  here  shown,  the  log  sin 
(for  departure)  and  log  cosine  (for  latitude)  may  be  at  once 
found  from  Table  IV.  If  these  logarithms  are  both  taken  out 
at  the  same  time  and  then  the  logarithms  of  the  distance  from 
Table  III.,  this  can  be  applied  to  both  log  sin  and  log  cos,  thus 
giving  the  log  departure  and  log  latitude,  when  from  Table  III. 
again  we  may  obtain  the  lat.  and  dep.  of  this  course,  giving 
these  their  signs  according  to  the  quadrant  in  which  the  azi- 
muth of  the  line  falls. 

If  Table  IV.  is  to  be  used  for  bearings  of  lines  as  given  by  a 
needle-compass,  then  enter  the  table  lor  the  given  bearing,  in 
the  first  set  of  angles,  beginning  at  o and  ending  at  90°. 

Example:  Compute  the  latitudes  and  departures  of  the  survey  plotted  in 
Fig.  55,  p.  185,  by  Tables  III.  and  IV.  The  following  are  ihe  field-notes  as  they 
would  appear,  first,  as  read  by  a transit  and  referred  to  the  true  meridian;  and, 
second,  as  read  by  a needle-compass: 


Station. 

Azimuth  referred  to 
the  South  Point. 

Compass  bearing. 

Distance. 

A 

290°  45' 

S.  69°  15'  E. 

7.06 

B 

217°  15' 

N.  37°  15'  E. 

5-93 

C 

140°  30' 

N.  39*"  30'  W. 

6.00 

D 

57"  45' 

S.  57°  45'  W. 

4-65 

E 

30°  00' 

S.  30°  00'  w. 

4.98 

SURVEYING. 


190 


The  following  is  a convenient  form  for  computing  the  lati- 
tudes and  departures: 


Course 

AH 

4th  Q. 

Course 

BC 

34  0. 

Course 
Cl) 
ad  g. 

Course 
DK 
isi  g. 

Course 

KA 

isi  g. 

log  sin  (dep.)  = 

9.9708 

9.7820 

9-8035 

^.C)2-I2 

9,6990 

log  dist.  = 

.8488 

.7731 

.7782 

.6675 

.6972 

log  dep.  = 

,8196 

.5551 

.5817 

•5947 

.3962 

Departure  = 

4-6.60 

+ 3.59 

— 3.82 

- 3-93 

- 2.49 

log  cos  (lat.)  = 

9-5494 

9.9009 

9.8874 

9-7272 

9-9375 

log  dist.  = 

.8488 

•7731 

.7782 

.6675 

.6972 

log  lat.  = 

.3982 

.6740 

.6656 

•3947 

•6347 

Latitude  = 

— 2.50 

4-4.72 

4-4-63 

— 2.48 

- 4-31 

It  is  seen  that  Table  IV.  answers  equally  well  for  either  set 
of  bearings,  and  also  that  Table  III.  would  have  given  the  lati- 
tudes and  departures  to  the  fourth  significant  figure  as  well  as 
to  the  third.  If  the  proper  quadrant  is  given  for  each  course 
in  the  heading  as  shown  above,  then  the  signs  may  be  at  once 
given  to  the  corresponding  latitudes  and  departures. 

174.  Balancing  the  Survey. — If  the  bearings  and  lengths 
of  all  the  courses  had  been  accurately*  determined,  the  survey 
would  “ close that  is,  when  the  courses  are  plotted  succes- 
sively to  any  scale  the  end  of  the  last  course  would  coincide 
on  the  plot  with  the  beginning  of  the  first  one.  Furthermore, 
the  sum  of  the  northings  (plus  latitudes)  would  exactly  equal 
the  sum  of  the  southings  (minus  latitudes),  and  the  sum  of  the 

* The  error  of  closure  simply  shows  a want  of  uniformity  of  measurement, 
for  if  all  the  sides  were  in  error  by  the  same  relative  amount,  the  survey  would 
close  just  the  same.  For  instance,  if  an  erroneous  length  of  chain  were  used, 
the  survey  might  close  but  the  area  be  considerably  in  error.  See  arts.  175 
and  177. 


LAND  SURVEYING. 


I9I 


eastings  (plus  departures)  would  exactly  equal  the  sum  of  the 
westings  (minus  departures).  It  is  evident  that  such  exactness 
is  not  attainable  in  practice,  and  that  neither  the  north  and 
south  latitudes  nor  the  east  and  west  departures  will  exactly 
balance,  there  always  being  a small  residual  in  each  case. 
These  residuals  are  called  the  errors  of  latitude  and  departure 
respectively.  The  distribution  of  these  errors  is  called  bal- 
ancing the  survey. 

In  the  form  for  reduction  of  the  field-notes  given  below, 
wherein  this  example  is  solved,  it  is  seen  that  the  error  of  lati- 
tude is  6 links  and  the  error  of  departure  is  5 links.  The  dis- 
tribution of  these  errors  is  made  by  one  of  the  following: 


FORM  FOR  COMPUTING  AREAS  FROM  BEARINGS  AND  DISTANCES 
OF  THE  SIDES. 


Sta- 

Courses. 

Dif. 

Lat. 

Departure. 

Balanced. 

Q 

+ 

tions. 

Bearings. 

1 

Dist. 

N. 

•f 

s. 

E. 

+ 

W. 

Lat. 

Dep. 

s' 

d 

Area. 

Area. 

A 

S.  69°  15'  E. 

Ch. 

7.06 

2.50 

6.60 

— 2.52 

-I-6.61 

6.61 

16.66 

B 

N.  37°  15'  E. 

5-93 

4.72 

3-59 

+ 4-71 

+ 3-6o 

16.82 

79.22 

C 

N.  39°  30'  W. 

6.00 

4-63 

3-82 

-(-4.62 

- 3-8i 

16.61 

76.74 

D 

S.  57°  45' W. 

4-65 

2.48 

3-93 

- 2.49 

— 3-92 

8.88 

22.11 

E 

S.  30®  00'  W. 

4.98 

4-31 

2.49 

- 4-32 

— 2.48 

2.48 

10.71 

28.62 

9-35 

9.29 

10.19 

10.24 

155-96 

49.48 

9.29 

10,19 

49.48 

Error  in  lat. 

= .06 

Error  in  dep. 

= .05 

106.48 

Error  of  closure  = 


0.0027 


Area  = 53-24  sq.  ch. 
= 5. 324^  erf s 


= I ia  366. 


2862 


192 


SUR  VE  YING. 


RULES  FOR  BALANCING  A SURVEY. 

Rule  i.  As  the  sum  of  all  the  distauces  is  to  each  particidar 
distance,  so  is  the  whole  error  in  latitude  {or  departure)  to  the  cor- 
rection of  the  corresponding  latitude  {or  departure),  each  correc- 
tion being  so  applied  as  to  diminish  the  whole  error  in  each 
case. 

Rule  2.  Determine  the  relative  difficulties  to  accurate 
measurement  and  alignment  of  the  several  courses,  selecting 
one  course  as  the  standard  of  reference.  Thus,  if  the  standard 
course  would  probably  give  rise  to  an  error  of  i,  determine 
what  the  errors  for  a7i  equal  distance  on  the  other  courses 
would  probably  be,  as  1^,2,  1,0.5  Multiply  the  length 

of  each  course  by  its  number,  or  weight,  as  thus  obtained. 
Then  we  would  have : 

As  the  sum  of  all  the  multiplied  lengths  is  to  each  multiplied 
length,  so  is  the  whole  error  in  latitude  {or  departure)  to  the  cor- 
rection of  the  corresponding  latitude  {or  departure),  each  correc- 
tion being  so  applied  as  to  diminish  the  whole  error  in  each 
case. 

These  two  rules  are  based  on  the  assumption  that  the  error 
of  closure  is  as  much  due  to  erroneous  bearings  as  to  erroneous 
chaining,*  which  experience  shows  to  be  true  in  needle-compass 
work. 

If,  however,  the  bearings  are  all  taken  from  a solar  compass 
(or  attachment)  in  good  adjustment,  or  if  the  exterior  lines  are 
run  as  a traverse  with  a transit,  so  that  the  angles  of  the  pe- 
rimeter are  accurately  measured,  then  the  above  assumption 
does  not  hold,  as  it  is  highly  probable  that  the  error  of  closure 
is  almost  wholly  due  to  erroneous  chaining.  Especially  would 
this  be  highly  probable  if  the  azimuth  is  checked  by  occupying 

* Let  the  student  prove  the  correctness  of  rules  i and  3 for  the  assumed 
sources  of  error. 


LAATD  SURVEYING. 


193 


the  first  station  on  closing  and  redetermining  the  azimuth  of  the 
first  course,  as  found  from  the  traverse,  and  comparing  it  with 
the  initial  (true  or  assumed)  azimuth  of  this  course.  If  it  thus 
appears  that  the  traverse  is  practically  correct  as  to  angular 
measurements,  it  may  be  fairly  assumed  that  the  error  of 
closure  is  almost  wholly  due  to  erroneous  chaining.  In  this 
case  use 

Rule  3.  As  the  arithmetical  sum  of  all  the  latitudes  is  to  any 
one  latitude,  so  is  the  zvhole  error  in  latitude  to  the  correction  to 
the  corresponding  latitude,  each  correction  being  so  applied  as 
to  diminish  the  whole  error  in  each  case.  Proceed  similarly 
with  the  departures.* 

In  the  solution  given  on  p.  191  the  first  rule  is  applied.  In 
ordinary  farm-surveying  it  is  not  common  to  give  the  lengths 
of  the  courses  nearer  than  the  nearest  even  link  or  hundredth 
of  a chain.  In  balancing,  therefore,  the  same  rule  may  be 
observed. 

175.  The  Error  of  Closure  is  the  ratio  to  the  whole  pe- 
rimeter of  the  length  of  the  line  joining  the  initial  and  final 
points,  as  found  from  the  field-notes.  The  length  of  this  line 
is  the  hypotenuse  of  a right  triangle  of  which  the  errors  in 
latitude  and  departure  are  the  two  sides.  Its  length  is  there- 
fore equal  to  the  square  root  of  the  sum  of  the  squares  of 
these  two  errors.  This  divided  by  the  whole  perimeter  gives 
the  error  of  closure,  which  ratio  is  usually  expressed  by  a 
vulgar  fraction  whose  numerator  is  one,  being  in  the 

above  example. 

The  error  of  closure  for  ordinary  rolling  country  should  not 


* It  is  evident  that  the  courses  could  here  be  weighted  for  different  degrees 
of  difficulty  in  the  chaining ; but  instead  of  multiplying  the  lengths  of  the 
courses  by  their  weights,  multiply  the  latitudes  and  departures  by  the  weights 
of  the  corresponding  courses,  and  then  distribute  the  errors  in  latitude  and 
departure  by  these  multiplied  latitudes  and  departures. 

13 


194 


SUR  VE  YING. 


be  more  than  i in  300.  In  city  work  it  sliould  be  less  than  i 
in  1000,  and  should  average  less  than  i in  5CXX). 

176.  The  Form  of  Reduction. — On  p.  191,  the  ordinary 
form  of  reduction  is  shown.  Here  the  courses  are  not  weight, 
ed  for  different  degrees  of  difficulty  in  chaining;  and  since  it 
was  a compass-survey  the  effect  of  erroneous  bearings  is  sup- 
posed to  equal  that  from  erroneous  chaining,  and  so  the  first 
rule  for  balancing  is  used.  The  balanced  latitudes  and  de- 
partures having  been  found,  the  double  meridian  distances  are 
next  taken  out.  In  taking  out  these  it  is  preferable  to  begin 
with  the  most  westerly  coriier^  whether  this  be  the  first  course 
recorded  or  not.  In  the  example  solved  on  p.  19 1,  it  is  the 
first  corner  occupied,  but  in  that  given  on  p.  198  it  is  not  the 
first  course.  By  beginning  with  the  most  westerly  corner 
(which  is  equivalent  to  passing  the  reference  meridian  through 
that  corner), all  the  double  meridian  distances  will  be  positive; 
otherwise  some  of  them  may  be  negative.  If  attention  be 
paid  to  signs  we  may  begin  at  any  corner  to  compute  the 
double  meridian  distances. 

A check  on  the  computation  of  the  D.  M.  D.’s  is  that,  when 
computed  continuously  in  either  direction  and  from  any  cor- 
ner, the  numerical  value  of  the  D.  M.  D.  of  the  last  course 
must  equal  its  departure.  This  is  a very  important  check  and 
must  not  be  neglected,  as  it  proves  the  accuracy  of  all  the  D. 
M.  D.’s. 

We  are  now  able  to  compute  the  double-areas  according  to 
equation  (2),  art.  171,  since  the  terms  entering  in  that  equation 
have  their  numerical  values  determined.  The  several  products, 
being  the  partial  double-areas,  are  written  in  the  last  two  col- 
umns, careful  attention  being  paid  to  the  signs  of  these  prod- 
ucts. Thus,  when  the  reference  meridian  is  taken  through  the 
most  westerly  corner,  then  all  the  D.  M.  D.’s  are  positive  and 
the  results  take  the  sign  of  the  corresponding  latitude.  If 
some  of  the  D.  M.  D.'s  are  negative,  then  the  signs  of  these  par- 


LAATD  SURVEYING. 


195 


tial  areas  are  opposite  to  those  of  the  corresponding  latitudes. 
The  algebraic  sum  of  the  partial  double-areas  is  twice  the 
area  of  the  figure,  as  shown  in  eq.  (2),  art.  171.  If  the  dis- 
tances are  given  in  chains,  then  the  area  is  given  in  sq. 
chains,  and  dividing  by  ten  gives  the  area  in  acres.  If  the  dis- 
tances were  given  in  feet,  as  it  often  is,  being  measured  by  a 
loo-foot  chain  or  tape,  then  the  area  is  in  sq.  feet,  and  this 
must  be  divided  by  43560,  the  number  of  sq.  feet  in  one  acre, 
to  give  the  area  in  acres.  This  is  best  done  by  logarithms,  as 
shown  in  the  example  solved  on  p.  198.  It  is  preferable  to  ex- 
press areas  in  acres  and  decimals  rather  than  in  roods  and 
perches,  as  was  formerly  the  custom. 

On  the  following  page  is  the  reduction  of  the  field-notes 
given  on  p.  183.  Here  the  several  courses  have  been  weighted 
for  various  degrees  of  difficulty  in  the  chaining.  Thus,  the  first 
and  second  courses  were  along  the  public  highway  and  on  even 
ground.  These  are  taken  as  the  standard  and  given  the 
weight  unity.  The  third  course  is  on  very  uneven  ground  and 
is  judged  to  give  rise  to  about  three  times  the  error  of  courses 
one  and  two  per  unit’s  distance.  It  is  therefore  weighted 
pthree.  The  proper  weight  to  give  to  the  several  courses  is 
thus  seen  to  depend  on  the  character  of  the  obstructions  to  ac- 
curate work,  and  represents  simply  the  judgment  of  the  sur- 
veyor as  to  the  probable  relation  of  these  sources  of  error. 
The  short  course  FG  was  very  difficult  to  measure,  as  there 
were  precipitous  bluffs,  and  the  course  GH  was  also  on  very 
uneven  ground. 

Following  the  column  of  weights  in  the  tabular  reduction 
are  the  multiplied  distances  ; the  errors  of  latitude  and  depart- 
ure are  distributed  according  to  the  results  in  this  column  by 
Rule  Two,  p.  192.  This  survey  was  also  made  with  a needle- 
compass. 

In  the  following  example  the  transit  was  used,  and  the 


96 


SU/^  VE  Y TNG. 


^ ' 


tji+ 


;z:+ 


ui 

iS 

^5.2 

SQ 


m o *-•  O i-i  M 

I I + + + I I I 

c<^ 


xrt  M 


CO 


'8 


i;  > boiifl 

f2'5'5.S 


CO  w w 


W W ^ ^ ^ ^ ^ 


Sd 

i-i 

m 

o 

O 

o 

O 

VO 

c 

in 

ir> 

CO 

u 

rt 

o 

O 

O' 

O 

O' 

W 

U) 

CO 

00 

'I- 

CO 

C/J 

cn 

C/5 

c/5’ 

<:muQwf^oiu 


a, 

V 

-o 

u 

O 

u 

W 


O N 
O -I- 

O'  O' 


+ 

*o- 

c^ 


ZAN’D  SURVEYIVG. 


197 


survey  began  at  A.  The  azimuth  of  the  line  AB  (Fig.  57) 
was  found  by  a solar  attachment, 
and  then  the  other  courses  ran  as 
a traverse,  the  horizontal  limb  of 
the  transit  being  oriented  by  the 
back  azimuth  of  the  last  course. 

The  azimuths  of  the  courses  are 
all  referred  to  the  south  point  as 
zero,  and  increase  in  the  direction 
SWNE.  After  the  last  course 
FA  was  run,  the  instrument  was 
carried  to  A and  oriented  by  a 
back  sight  on  i^and  the  azimuth 
of  AB  again  determined.  This 
agreed  so  well  with  the  original 
azimuth  of  this  course  that  the 
azimuths  of  all  the  courses  were 
proved  to  be  correct.f 

The  error  of  closure  is  therefore  due  to  the  chaining  alone. 
A hundred-foot  chain  was  used  so  that  the  distances  are  all 
given  in  feet.  The  obstructions  to  chaining  were  about  uni- 
form, so  the  courses  are  all  given  equal  weight.  In  balancing. 
Rule  Three  must  be  used,  since  the  errors  are  supposed  to 
come  only  from  the  chaining. 

If  the  errors  in  latitude  and  departure  had  been  distributed 
by  Rule  One,  or  in  proportion  to  the  lengths  of  the  courses, 
the  resulting  area  would  have  been  56.41  acres,  a difference  of 
0.07  acres,  or  about  one  eight-hundredth  of  the  total  area. 

177.  Area  Correction  due  to  Erroneous  Length  of 


*The  lines  MB  and  00'  in  this  figure  are  used  in  art.  186. 
f From  the  azimuth  check  here  obtained,  as  compared  to  the  errors  in  lat- 
itude and  departure,  decide  whether  the  latter  are  due  mostly  to  the  chaining 
or  whether  the  errors  in  azimuth  have  had  an  equal  influence,  and  so  determine 
whether  to  use  rule  i or  rule  3 in  balancing. 


198 


SURVEYING. 


Areas. 

• • • 00  0 

• * * CO  GO  * 

• • • \n  0 • 

I r ; CO  CO  r 

. . . CT' 

. CO 

CO 

0 

vO 

CO 

+ 

Areas. 

•t"  0 N • . vn 

CO  CO  . .CO 

O-  O- 

o'  c4  cT  t * »A 

-t-  00  0 , • 

0 0-  : ; 

ci  ci  • • 

• . 

• . 

M CO 

O'  vO 

0 vO 

vo"  o‘ 

CO  C7' 

CO  CO 

VO 

j D.  M.  D. 

O'  0 CO  -t  N 10 

CO  -1-  CO  »0  CO 

VO  r>  CO  r'.  CO 

N CX  CX  fi 

Balanced. 

d 

<u 

Q \ 

►-  M -f  0 N VO 

CO  CO  CO  VO  CO 

ex  O-  CO  CO 

1 + 1 1 1 + 

rt 

0 0 0 M 0 

0 0 CO  -1- 

00  O'  CO  VO  VO 

+ + 4-  1 T + 

Departure. 

' 

0 • 0 w 

CO  • CO  *7- 
N • CO 

O' 

0 

CO 

w+ 

• O"  • • • M 

. to  • • ‘O' 

. r:J-  ...  to 

0 O' 

CX  0 

CO  CO 

Dif.  Lat. 

t/j  1 

• • • cn  Cl  • 

• • • w ^ • 

• • • m • 

VO  O' 

0 VO  1 

0 0 1 

ex  cx 

VO  VO  CO  . . M 

0 0 CO  • • M 

CO  O'  CO  . • 

• » 

O' 

VO 

0 

ex 

Dist. 

838  ft. 

1004 

896 

912 

1542 

1392 

•cl- 

CO 

Azimuth. 

! 

1 

"cO  o 0 Q Tj-  CX 

0 CO  M 5 0 CO 

0 

rt-  ir>  VO  0 O' 

0 0 *H  VO  0 

M « M M 

A (check)  164°  05'  | 

Station. 

pq  u Q W 

CO 


O 

o' 


•ct 


a. 

o 

*0 


o 

II 


o 

U 


4/6‘J  4-  172  ( 2,470,012  sq.  ft. 

Error  of  closure  = • = i in  360.  (435^0  sq.  ft.  = lA.)  Area  = 

6484  tor  56.70  acres. 


ZAArn  SURVEYING. 


199 


Chain. — If  the  measuring  unit  has  not  the  length  assigned  to 
it  in  the  computation,  then  the  computed  area  will  be  errone- 
ous. Such  an  error  will  not  show  in  the  balancing  of  the  work 
or  elsewhere,  and  hence  an  independent  correction  must  be  ap- 
plied for  this  error.  If  the  chain  was  too  long  by  one  one- 
thousandth  part  of  its  length,  for  instance,  then  all  the  courses 
are  too  short  in  the  same  ratio.  And  since  similar  plane  fig- 
ures are  to  each  other  as  the  squares  of  their  like  parts,  we 
would  have 

true  area  : computed  area  ::  (1001)“  : (1000)*, 

or  true  area  = computed  area  (nearly)  ;* 

or,  in  general,  if  / = length  of  chain  and  Al  = error  in  length, 
being  positive  for  chain  long  and  negative  for  chain  short,  and 
if  Al  is  small  as  compared  with  /,  as  it  always  is  in  this  case, 
then  if  we  let 

A = true  area.  A'  = computed  area 
Ca  — correction  to  computed  area, 
and  A = relative  error  of  chain, 

IA-2AI  ^ 

we  have  A = — A' — {i 2A')A' 

whence,  A — A'  = Cj_=  2 A A'. 

That  is  to  say,  the  relative  area  correction  due  to  erroneous 
length  of  chain  is  twice  the  relative  error  of  the  chain^  being 
positive  for  chain  long,  and  negative  for  chain  short. 


* The  error  in  this  approximation  is  one  one-millionth  in  this  case,  and 
would  always  be  inconsiderable  in  this  class  of  problems. 


200 


S UR  VE  YING. 


FINDING  TH?:  AREA  OR  SUPERFICIAL  CONTENTS  OF  LAND 
WHEN  THE  RECTANGULAR  COORDINATES  OF  THE  COR- 
NERS ARE  GIVEN  WITH  RESPECT  TO  ANY  POINT  AS  AN 
ORIGIN. 

178.  Conditions  of  Application  of  this  Method. — 

Where  many  tracts  of  land,  all  bounded  by  straight  lines,  are 
somewhat  confusedly  intermingled,  as  is  the  case  in  many  of 
the  older  States,  and  where  the  area  of  each  tract  over  an  ex- 
tended territory  is  to  be  found,  this  method  is  greatly  to  be 
preferred  to  that  by  means  of  the  boundary-lines.  In  this  case 
it  is  only  necessary  to  make  a general  coordinate  survey  of  the 
whole  territory,  as  described  in  Chapter  VIII.,  on  Topographi- 
cal Surveying,  using  the  stadia  for  obtaining  distances,  and  be- 
ing careful  to  locate  every  corner  of  each  tract.  If  areas  alone 
are  required,  no  attention  need  be  paid  to  the  obtaining  of 
elevations  for  contour  lines,  and  so  the  work  is  greatly  facilitated. 
A transit  and  two  or  three  stadia  rods  would  be  the  instru- 
ments used.  The  survey  would  then  be  carefully  plotted  and 
the  coordinates  measured  on  the  sheet,  or  they  could  be  com- 
puted from  the  field-notes.  If  the  plotting  is  carefully  done 
the  former  method  is  preferable.  It  is  best  to  choose  the 
origin  of  coordinates  entirely  outside  the  tract  and  so  that  the 
whole  area  falls  in  one  quadrant,  thus  making  all  the  coor- 
dinates of  one  sign. 

Large  tracts  of  mineral  land  are  sometimes  acquired  by 
large  companies,  including  perhaps  hundreds  of  individual  es- 
tates. In  such  cases  a topographical  map  of  the  region  is 
necessary ; and  when  this  survey  is  rnade,  a little  extra  care  to 
obtain  all  the  “ corners”  of  private  claims  will  enable  the  areas 
of  all  suclrlots  to  be  determined  with  great  accuracy  and  at 
small  additional  cost.  The  method  probably  has  no  advan- 
tages when  the  area  of  but  a single  tract  is  desired. 


LAATD  SUJ^VEVING. 


201 


179.  The  Method  of  Finding  the  Area  from  the  Rec- 
tangular Coordinates  of  the  Corners  is  as  follows  : 

Let  Fig.  58  be  the  same  tract  as  that  given  in  Fig.  55,  and 


Fig.  58. 


let  the  origin  be  one  chain  west  of  A and  three  chains  south  of 
B.  Then,  from  the  balanced  latitudes  and  departures  for  this 
case,  given  on  p.  191,  we  find  the  following  coordinates  of  the 
corners  fb,  etc.,  denoting  the  latitudes  of  the  corners  A,  B, 
etc.,  and  similarly  with  Xb,  etc.,  for  departures : 


= 5-52,  n = 3-00,  = 77L  n = 12.33,  7,  = 9.84. 

.^^=1.00,  Xb  = 7.6i,  Xc=  11.21,  xa  = 7.40,  = 3.48. 

The  area  of  the  figure  ABODE  is  equal  to  the  areas 


ybBCy^  -^-y^CDy^  — {y^EDy^  y^AEy,  + y,,BAyJi ; 


202 


SURVEYING. 


or 

^ i [(jj'c-jz,)  (^6+^c)  + (^d-7c)  (^0  + x^-{ya  -y,)  (xa  + x,) 
- {ye  — ya)  (^e  + -^a)  “ ( Ja  “ /b)  {^a  + ^b)]-  ( I ) 

. By  developing  equation  (i)  we  obtain 
A = i [y^Xc  — yaXt,  +ybXa  — JbXc  + “ /c^<l 

+ yaXc  — ydXe  + yeXd  — /e^a]  • (2) 

From  this  we  may  obtain  either  of  the  following: 


A =i[ya  (Xe  — Xb)  +^6  (^a  — +^o  (Xb  — Xa) 

+ yi  {Xc  - X,)  +ye  {Xa  - x^l  ; 
or 

A--l\Xa  {ye  — yb)  + Xb  (ya  —ye)  + Xe  (yb  - ya) 

+ {ye  - ye)  +Xe{yd-  ya)}-  , 


(3) 


From  these  equations  we  may  obtain  the  following 


RULE  FOR  FINDING  THE  AREA  OF  A CLOSED  FIGURE 
BOUNDED  BY  STRAIGHT  LINES  FROM  THE  RECTANGULAR 
COORDINATES  OF  THE  CORNERS. 


Multiply  the 


abscissa 

ordinate 


tween  the 


( ordinates 
1 abscisses 


j-  to  each  corner  by  the  difference  be- 
of  the  two  adjacent  corners,  always  making 


the  subtraction  in  the  same  direction  around  the  figure,  and  take 
half  the  sum  of  the  products. 

The  student  will  observe  that  this  is  simply  a more  general 
case  of  the  former  method  of  computing  the  area  from  the 
latitudes  and  double-meridian  distances. 


LAND  SUDVEY/NG. 


203 


180.  The  Form  of  Reduction  for  this  case  is  given  below. 


Corner. 

Ordinates 

Abscissae 

W. 

Difference 
between  Alter- 
nate Abscissae. 

Double  Areas. 

A 

5-52 

1. 00 

- 4.13 

— 

22.80 

B 

3.00 

7.61 

— 10.21 

— 

30.63 

C 

7.71 

II. 21 

+ .21 

+ 

1.62 

D 

12.33 

7.40 

+ 7-73 

+ 

95.31 

E 

9.84 

3.48 

+ 6.40 

62.98 

Plus  areas  = 159.91 
Minus  areas  = 53-43 
2 ) 106.48 

Area  = 53.24  sq.chns, 
= 5 . 324  acres. 

This  is  the  same  result  as  found  on  p.  191  by  the  other 
method,  as  it  should  be,  since  the  same  balanced  latitudes  and 
departures  were  used  in  each  case.  ' 

It  is  also  evident  that  after  the  balanced  latitudes  and 
departures  are  obtained  for  the  ordinary  perimeter-survey,  the 
area  may  be  computed  by  this  form — from  equations  (3),  p. 
202,  if  preferred.  Or,  if  the  coordinates  of  the  corners  are 
taken  at  once  from  a map,  or  computed  from  traverse  lines, 
the  bearings  and  lengths  of  the  courses  joining  such  corners 
could  readily  be  computed.  Thus,  the  length  of  any  course, 
as  BC,  is  BC  = V -f-  (/„  — >'6)^  while  its  bearing  is 

, , . ^c  — 

the  arc  whose  tan  is . 

yc  -yb 

181.  Supplying  Missing  or  Erroneous  Data. — In  any 
closed  survey  there  are  two  geometric  conditions  that  must 
be  fulfilled,  viz.  : 

1.  The  sum  of  all  the  latitudes  must  be  zero. 

2.  The  sum  of  all  the  departures  must  be  zero. 


204 


SU/^  VE  Y/NG. 


These  two  conditions  give  rise  to  two  corresponding  equa- 
tions. 

If  /j,  4,  /g,  etc.,  be  the  lengths  of  the  several  courses,  and  if 
etc.,  be  their  compass-bearings,  then  our  two  geo- 
metric conditions  give 

/,  sin  6^  -(-  4 sin  -f-  4 sin  0^  -j-  etc.,  =0.  . . (i) 

/,  cos  + 4 cos  + ^3  cos  <^3  + etc.,  = o.  . . (2) 

Since  we  have  two  independent  equations,  we  can  solve  for 
two  unknown  quantities.  These  two  unknowns  may  be  any 
two  of  the  functions  entering  in  the  above  equations.  Thus, 

if  any  two  distances,  any  two  bearings,  or  any  one  distance 

and  any  one  bearing  are  missing,  they  may  be  found  from 
these  equations.  Or,  if  but  one  bearing  or  distance  is  missing, 
it  may  be  found  from  one  of  these  equations  and  the  other 
equation  used  for  balancing  either  the  latitudes  or  departures. 
When  all  bearings  and  distances  are  given,  these  equations  are 
really  used  in  balancing  ; but  if  they  are  both  used  to  deter- 
mine missing  quantities,  there  can  be  no  balancing  of  errors, 
for  when  the  missing  quantities  are  computed  by  these  equa- 
tions, both  latitudes  and  departures  will  exactly  balance.  In 
other  words,  all  the  errors  of  the  survey  are  thus  thrown  into 
these  two  quantities. 

This  artifice  should  therefore  never  be  resorted  to  except 
where  it  is  impracticable  to  actually  measure  the  quantities 
themselves  in  the  field. 

There  are  four  cases  to  be  solved : 

I.  Where  the  bearing  and  length  of  one  course  are  un- 
known. 

II.  Where  the  bearing  of  one  course  and  length  of  another 

o O 

are  unknown. 

III.  Where  two  bearings  are  unknown. 

IV.  Where  two  lengths  are  unknown. 


LAND  SURVEYING, 


205 


The  bearings  will  be  reckoned  from  both  north  and  south 
points  around  to  the  east  and  west  points,  as  is  common  in 
compass  surveying.  Then  the  length  of  a course  into  the  sin 
of  its  bearing  gives  its  departure,  and  into  the  cos  of  its  bear- 
ing  gives  its  latitude.  North  latitude  is  plus  and  south  latitude 
minus;  east  departure  plus  and  west  departure  minus. 

In  every  case  let  the  sum  of  the  departures  of  all  known 
courses,  taken  with  the  opposite  sign,  be  D,  and  the  sum  of 
their  latitudes,  taken  with  the  opposite  sign,  be  L.  Then  D 
and  L are  the  departure  and  latitude  necessary  to  close  the 
survey. 

Case  I. — Bearing  and  le^igth  of  one  course  unknown. 

The  two  condition  equations  here  become 


(3) 


Whence 


tan 


(4) 


Having  found  the  bearing,  find  4i  from  either  of  equations 
(3).  Particular  attention  must  always  be  paid  to  the  signs  of 
D and  L.  Evidently  sin  6^  (dep.)  and  cos  (lat.)  have  the 
same  signs  as  D and  L respectively,  whence  the  quadrant 
which  includes  the  bearing  may  be  determined  and  the  proper 
letters  applied.  For  this  purpose  Fig.  56  may  be  consulted. 

Case  1 1. — The  bearing  of  one  course  and  the  length  of  another 
unknoivn. 

In  this  case  let  a be  the  known  bearing  of  the  course  whose 
length  is  unknown,  and  let  / be  the  known  length  of  the  course 
whose  bearing  is  unknown.  Then  we  have 


s\n  a-\- 1 sin  6^  = L 
cos  a-\- 1 cos  6^  = L 


2o6 


SUR  VE  Y I NG. 


If  we  let  sin  a — s^  and  cos  a = Cy  we  have 


L = sD+cL±  Vr  - {D^  + U)  + {sD  + cL)\  . (6) 


Here  there  are  two  values  of  which  will  satisfy  the  equa- 
tion, and  so  there  arc  two  solutions  to  the  problem.  If  the 
surveyor  has  no  knowledge  whatever  of  either  the  unknown 
length  or  bearing,  the  problem  is  indeterminate.  If  he  has 
seen  the  tract  he  could  usually  tell  which  length  or  which 
resulting  bearing  was  the  correct  one,  when  the  problem  would 
become  determinate.  When  4^  is  found,  substitute  in  one  of 
equations  (5)  and  find  6^.  Pay  careful  attention  to  the  signs 
of  the  trigonometrical  functions  of  all  bearings. 

Case  III. — When  two  bearuigs  are  unknown. 

Let  I'  and  1"  be  the  known  lengths  of  the  courses  whose 
bearings  are  unknown.  Then  the  equations  become 


I'  sin  -|-  /"  sin  6^  = D;) 

I cos  -[■  cos  6n  = L.  \ ‘ 


• • • (7) 


Whence 


Where 


cos  6^  = 


KL  ± DVD  - 
D^  + D 


• (8) 


K = 


^ p D' D 
2l" 


This  case  is  also  indeterminate  unless  one  is  able  to  tell 
which  of  the  two  sets  of  bearings  is  the  correct  one. 

Case  IV. — When  the  lengths  of  two  courses  are  unknown. 
Let  a and  b be  the  known  bearing  of  the  courses  whose 
lengths  are  unknown. 


ZAJVn  SURVEYING. 


207 


Our  equations  here  become 


whence 


sin  a-{-  Insm  b = D\ 
cos  a-\-  In  cos  b = L. 

Ls,m  a — D cos  a 
” sin  {a  — b) 


• • (9) 


(10) 


This  case  is  determinate. 

In  case  there  is  but  one  unknown,  then  either  one  of  equa- 
tions (3)  will  solve.  In  taking  out  either  the  sine  or  the  cosine 
from  the  tables,  however,  two  angles  will  always  be  found 
equidistant  from  the  east  or  west  point  if  the  sine,  and  equi- 
distant from  either  the  north  or  south  point  if  the  cosine, 
either  of  which  may  be  chosen.  In  such  case  both  sine  and 
cosine  must  be  found,  when  the  signs  alone  of  these  two  func- 
tions will  determine  the  quadrant  in  which  the  bearing  is  found. 
Hence,  if  the  single  unknown  is  a bearing;  both  of  the  equa- 
tions (3)  must  be  used  in  order  to  determine  which  of  the  two 
bearings  given  by  the  table  is  the  correct  one,  but  one  alone  is 
sufficient  to  obtain  the  numerical  value  of  the  bearing.  Thus, 
if  the  sine  equation  is  used  to  compute  the  bearing,  then  the 
latitude  may  be  taken  out  for  the  given  length  and  bearing; 
and  these  will  then  not  balance,  but  will  have  to  be  balanced 
in  the  usual  way,  while  the  departures  will,  of  course,  balance, 
since  the  residual  departure  D necessary  to  close  the  survey  as 
to  departures  was  used  to  compute  the  corresponding  bearing. 
The  reverse  of  this  would  be  true  if  the  cosine  equation  were 
used  to  compute  the  bearing. 


2o8 


S UR  VE  YING. 


PIXDTTING  THE  FIELD-NOTES. 

iSia.  To  plot  a Compass  Survey  select  a point  for  the 
initial  station,  and  pass  a meridian  through  it  in  pencil,  l^y 
means  of  a semicircular  protractor,  sucli  as  is  shown  in  Fig. 
44,  mark  the  bearing  and  draw  an  indefinite  line  from  the  sta- 
tion point.  On  this  line  lay  off  to  scale  the  length  of  the 
course,  thus  establishing  the  next  corner.  Througli  this  draw 
another  pencil  meridian,  and  proceed  as  before.  If  the  plot- 
ting is  perfect  the  length  of  the  line  joining  the  final  with  the 
initial  point,  taken  to  scale,  is  the  error  of  closure  of  the  sur- 
vey; and  the  horizontal  and  vertical  components  of  this  line, 
taken  to  scale,  should  be  the  errors  in  departure  and  latitude 
respectively  as  obtained  by  the  computation. 

If  preferred,  the  bearings  of  the  successive  courses  may  be 
so  combined  as  to  give  the  deflection-angle  at  each  station,  and 
these  laid  off  from  the  preceding  course  as  already  drawn. 
Errors  are  more  likely  to  accumulate  in  the  plot  by  this 
method,  however,  than  by  that  first  given. 

Again,  the  rectangular  coordinates  of  the  several  corners 
may  be  computed  and  these  plotted  from  a pair  of  rectangular 
axes,  but  this  is  not  a common  practice. 

For  the  plotting  of  transit  surveys,  especially  where  the 
stadia  is  used,  see  Chapter  VIII. 


THE  AREAS  OF  FIGURES  BOUNDED  BY  CURVED  OR  IRREGULAR 

LINES. 

182.  The  Method  by  Offsets  at  Irregular  Intervals. 

— Where  a tract  of  land  is  bounded  by  a body  of  water,  as  a 
stream  or  lake,  it  is  customary  to  run  straight  lines  as  near  the 
boundary  as  practicable  and  then  to  take  rectangular  offsets 
at  selected  intervals  from  these  bordering-lines  to  the  irregular 
boundary.  These  small  areas  are  then  computed  as  trapezoids, 


LAND  SURVEYING. 


209 


the  distance  along  the  base-line  being  the  altitude  and  the  half- 
sum of  the  adjacent  offsets  being  the  mean  width.  The  offsets 
should  therefore  be  run  at  such  intervals  as  to  make  this 
method  of  computation  sufficiently  accurate.  Such  offsets 
were  taken  from  the  course  GH  in  Fig.  54,  the  notes  for  which 
are  given  on  p.  183. 

The  work  of  computation  may  be  shortened  by  using  a 
modified  form  of  the  method 
of  areas  from  the  rectangular 
coordinates  of  the  corners,  b 
which,  in  this  case,  are  the  ends  ” 
of  the  offset  lines.  Let  Fig. 

59  be  an  area  to  be  determined 
from  the  offsets  from  the  line 
AK.  The  position  and  lengh  of  the  offsets  are  given.  Take 
the  origin  at  A and  let  the  distances  along  AK  be  the  abscissae, 
and  the  lengths  of  the  offsets  be  the  ordinates.  Using  the 
second  of  equations  (3),  p.  202,  we  have 


D 

F 

i 

1 

Vp 

1 

! 

1 

1 

^ K 

Si 

S! 

o5,i 

1 

1 

j 

Oil 

i 

1 

• 

• 1 

"1 

1 

^.1 

1 

1.00 

V 

0 1.21 

2JJ3 

3.56 

5.04. 

5.75 

7.00 

Fig. 


A — *k\Xa{yk  — J'i.)  + —yc)  + ^e{yb  —yd) 

+ ^d{y<,-  y.)  + - yf) 

+ ^Ay.-yg)  + Xg  {yf  -y^ 

+ (0 

But  here  Xi,,  ya,  and  y^g  are  all  zero  ; also  x^  = Xjg^  hence 
this  equation  becomes 

A=i[Xg{yi-  ya)  -f-x^iy,  — y.)  + ^e  {yd-  yf) 

+Mye-yg)+-Vg  {y/-yi,)+  Myg+yh)l(2) 

From  eq.  (2)  we  have  the 


* The  plus  sign  is  here  used,  since  we  have  gone  around  the  figure  in  a direc- 
tion opposite  to  that  followed  in  the  general  case 


210 


SURVEYING. 


RULE  FOR  FINDING  AREAS  FROM  RECTANGULAR  OFFSETS  AT 
IRREGULAR  INTERVALS. 

Multiply  the  distajice  along  the  course  of  each  intermediate 
offset  from  the  first  by  the  difference  between  the  two  adjace7it 
offsets^  always  subtractuig  the  following  from  the  preceding. 
Also  midtiply  the  distance  of  the  last  offset  from  the  first  by  the 
smn  of  the  last  two  offsets.  Divide  the  sum  of  these  products  by 
two. 

The  following  is  the  nunnerical  reduction  for  finding  the 
area  of  the  irregular  tract  shown  in  Fig.  59. 


Offset. 

Distance 
from  A. 

Length  of 
Offset. 

Differences. 

Products. 

ch. 

ch. 

ch. 

sq.  ch. 

B 

0.00 

1-53 

C 

I. 21 

1.76 

- 

0.47 

— 0.57 

D 

2.23 

2.00 

- 

0.56 

- 1.25 

E 

3-56 

2.32 

+ 

.09 

+ .32 

F 

5.04 

I. 91 

+ 

.87 

+ 4.38 

G 

5.75 

1-45 

.91 

+ 5-23 

H 

7.00 

1. 00 

+ 

2.45 

+17.15 

2 ) 25.26 


Area  = 12.63  sq.  chs. 

= 1 . 263  acres. 

It  is  evident  that  an  area  bounded  on  all  sides  by  irregular 
or  curved  lines  could  have  a base-line  run  through  it,  and  off- 
sets taken  from  this  line  to  both  boundaries  and  the  area  com- 
puted by  this  method.  Example  196,  p.  221,  should  be  so 
computed. 

183.  The  Method  by  Offsets  at  Regular  Intervals. — If 

the  intervals  between  the  offsets,  or  ordinates,  are  all  equal  the 
computation  is  much  simplified.  On  the'assumption  that  the 
area  is  a series  of  trapezoids,  we  have  the 


lAATB  SURVEYING. 


2II 


RULE  FOR  FINDING  THE  AREA  FROM  RECTANGULAR  OFFSETS 
AT  REGULAR  INTERVALS. 

Add  together  all  the  intermediate  offsets  and  one  half  the  end 
offsets^  and  multiply  the  sum  by  the  constant  interval  between 
them. 

The  following  rules  for  finding  areas  are  found  from  the  suc- 
cessive orders  of  differences  in  each  case  and  may  all  be  derived 
by  a rigid  development.*  They  assume  that  the  bounding-line 
is  curved  and  that  rectangular  ordinates  have  been  measured 
at  uniform  intervals  from  a base-line  traversing  the  figure. 

Let  the  common  interval  between  ordinates  be  d\  let  the 
lengths  of  the  ordinates  be  h^,  h^,  h^  . , , , hn  \ and  let  the 
number  of  intervals  be  N, 


1.  N = 1,  A = ^{h^-{-  hi),  Trapezoidal  Rule. 

N = 2,  A = ^ {h^  + Simpson’s  ^ Rule. 

'id 

III.  iV  = 3,  ^ — -g-  {K+  ZK  + iK  + ^0’  Simpson’s  f Rule. 


IV.  = 4,  ^ ^ [7  (/^o  + h)  + 32  {K  + h)  + I2/.J. 

V.  iv  = 6,  A IK  ■VK  + K+  K+  5 {K+  K+  K)+Kl 


This  is  called  Weddel’s  Rule.  If  a quadrant  be  computed  by 
this  rule,  the  result  is  o.yjgA  instead  of  o.785r^  the  true  value. 
When  an  area,  bounded  by  a base  line  and  two  end  ordi- 


*See  appendix  C. 


212 


SURVEYING. 


nates,  be  divided  by  imaginary  lines  parallel  to  the  end  ordi- 
nates and  equally  spaced,  as  in  Fig.  6o,  and  if  the  middle  ordi- 


Fig.  6o. 


nates  of  these  partial  areas  be  measured,  then  if  ^ common 
width  of  the  partial  areas  and  /i^,  etc.,  their  middle  ordi- 

nates, a the  first  end  ordinate  and  b the  last  one,  we  have, 
approximately. 


I.  ^ = d:2h, 


where  signifies  the  summation  of  all  the  h's. 

The  following  rules  are,  however,  more  accurate : 


II.  A = d:2h  + - ^ Poncelet’s  Rule ; 


or, 


[Rule. 


III.  A = + Francke’s 

/2 

The  various  rules  above  given  are  often  used  to  determine 
areas  of  irregular  figures  such  as  steam  diagrams,  cross-sections 
of  structural  forms,  streams,  excavations,  etc.  The  most 
ready  and  accurate  means  of  determining  all  such  areas,  how- 
ever, is  by  means  of  the  planimeter. 


LAJVB  SURVEYING. 


213 


THE  SUBDIVISION  OF  LAND. 

184.  The  Problems  arising  in  the  subdivision  of  land  are 
of  almost  infinite  variety.  All  such  problems  are  solved  by  the 
application  of  the  fundamental  principles  and  relations  of 
geometry  and  trigonometry  with  which  the  student  is  supposed 
to  be  familiar.  There  are,  however,  two  classes  of  problems  of 
such  frequent  application  that  they  will  be  given  in  detail. 

185.  To  cut  off  from  a Given  Tract  of  Land  a Given 
Area  by  a Right  Line,  starting  from  a Given  Point  in  the 
Boundary. — In  Fig.  55,  p.  185,  let  O be  the  middle  point  on 
the  line  AB,  from  which  a line  is  to  be  run  in  such  a manner 
as  to  cut  off  three  acres  from  the  western  portion  of  the  tract. 
We  may  at  once  assume  that  the  dividing-line  will  cut  the  side 
DC  in  some  point  X,  whose  distance  from  D is  to  be  found. 
First  compute  the  area  OAED,  using  the  balanced  latitudes 
and  departures  given  on  p.  191,  we  have  the  following: 


Course. 

Lat. 

Dep. 

D.  M.  D. 

Double  Areas. 

+ 

- 

AO 

ch. 

— 1.26 

ch. 

+ 3.30 

3.30 

4.16 

OD 

(+  8.07) 

(+  3-10) 

9.70 

78.28 

DE 

- 2.49 

- 3.92 

8.88 

22.11 

EA 

- 4.32 

— 2.48 

2.48 

10.71 

(—  8.07)  (—  3.10)  Sums  + 78.28  36.98. 

— 36.98 

2)41.30  » 


Area  = 20.65  sq.  chs. 

= 2.065  acres. 

Here  the  latitude  and  departure  of  the  course  OD  are  such 
as  to  make  the  latitudes  and  departures  balance.  The  area  is 


214 


SURVEYING. 


found  to  be  2.065  acres,  leaving  0.935  acres  to  be  laid  off  from 
OD  by  the  line  OX.  It  remains  now  to  find  the  point  X. 

First  compute  the  length  and  bearing  of  the  line  OD  from 
Case  I.,  p.  205. 

Thus  we  have 

D +3.10 

Whence  6—  21°  from  the  table  of  natural  tangents.  From 
the  table  of  natural  sines,  we  find  sin  21°  = 0.358. 

Hence  from  eq.  (3),  p.  205,  we  have 

/ sin  ^ = Z?,  or  0.358/ = 3.10. 

Whence  I — 8.66  chains. 

The  bearing  is  evidently  N.  21°  E. 

We  now  have  to  find  the  distance  DX  such  that  the  area 
ODX  shall  be  9.35  sq.  chains.  Since  the  area  of  any  triangle 
is  one  half  the  product  of  two  sides  into  the  sine  of  the  in- 
cluded angle  (another  way  of  saying  it  is  equal  to  half  the  base 
into  the  altitude),  we  have 

9.35  =1(8.66  sin  . . . . (i) 

From  the  bearings  of  OD  and  DX  we  find  the  angle  ODX 
to  be  bo""  30',  hence  sin  ODX  = 0.870,  from  which  we  find 

DX  = 2.48  chains. 

The  length  and  bearing  of  the  line  OX  may  be  computed 
from  its  latitude  and  departure,  the  same  as  was  done  for  the 
line  OD  above,  or  we  may  compute  the  angle  DOX  and  length 


LJATB  SURVEYING. 


215 


OX  by  solving  the  triangle  DOX.  The  bearing  of  OX  may 
then  be  found,  and  the  line  run  from  O.  There  will  then  be 
two  checks  on  the  work,  viz. : the  measured  lengths  of  OX  and 
DX  must  be  equal  to  their  computed  values. 

To  find  the  angle  DOX,  let  the  three  angles  of  the  triangle 
be  D,  O,  and  X,  and  the  sides  opposite  these  angles  be  d,  0, 
and  X,  respectively.  Then  we  have 

tanHX-£?)  = ^tani(^+  0- 

This  equation  gives  the  angle  (X  — O'),  whence 
0 = 1 (X+  £?)- J {X-  6'),and  ^ (X  + O)  + i (X  - O). 

Also,  d = OX  = OB 

Sin  X 

and  0 BX  = OB 

sm  X 

We  therefore  have  the  following 

RULE  FOR  CUTTING  OFF  A GIVEN  AREA  BY  A LINE  START 
ING  FROM  A GIVEN  POINT  IN  THE  BOUNDARY 

Having  first  surveyed  the  tract  and  plotted  the  same,  join 
the  given  point  on  the  plot  with  the  corner  which  will  give  the 
nearest  approximation  to  the  desired  area.  Compute  the 
length  and  bearing  of  this  line,  and  of  the  area  thus  cut  off. 
Subtract  this  area  from  the  desired  area,  and  the  remainder  is 
the  area  to  be  cut  off  in  the  form  of  a triangle,  one  side  of 
which  has  bearing  and  distance  given,  and  another  side  has  its 
bearing  alone  given.  From  these  data  compute  the  lengths  and 
bearings  of  the  other  sides,  one  of  which  is  the  line  sought. 
This  line  may  then  be  run,  and  its  length  measured,  as  well  as 
the  length  of  the  portion  of  the  opposite  boundary  cut  off,  for 
a check  on  the  accuracy  of  the  work. 

i86.  To  cut  off  from  a Given  Tract  of  Land  a Given 
Area  by  a Right  Line  running  in  a Given  Direction. 


2i6 


SUR  VE  YING. 


— Let  the  problem  be  to  cut  off  30  acres  from  the  northern 
portion  of  the  tract  shown  in  Fig.  57,  p.  197,  by  a line  whose 
bearing  is  N.  80°  E.,  or  whose  azimuth  is  260°.* 

Pass  a line  parallel  to  the  required  line  through  the  corner 
nearest  to  the  probable  position  of  the  desired  line.  Let  MB, 
Fig-  57>  such  a line.  Compute  the  lengths  of  the  lines  EM 
and  MB  by  Case  IV.,  p.  206. 

From  the  computation,  p.  198,  we  have  the  following: 


Courses 

Azimuth. 

Lengths. 

Ralanccd 

Latitudes. 

Balanced 

Departures. 

D.  M.  D.’s 

Double  Areas. 

BC 

205°  39' 

1004  ft. 

-j-  906  ft. 

+ 432  ft. 

2738 

-f-  2,480,628 

CD 

II2  12 

896 

+ 339 

- 834 

2336 

+ 

791,804 

DE 

55  00 

912 

— 522 

- 750 

752 

— 

392,544 

EM 

0 04 

(926) 

— 926 

— I 

I 

- 

926 

MB 

260  00 

(1171) 

+ 203 

+ 1153 

1153 

+ 

234.059 

(4- 723)  (-1152)  2)3,113,021 


Therefore  to  close  requires  Z = — 723  and  Z>  = -}-  1152.  Area  = 1,556,510 

sq.  ft. 

, ■ = 35.73  ac’s. 

From  equation  (10),  p.  207,  we  have 


EM^ 


D cos  260°  — L sin  260° 
sin  259°  56'  ‘ 


(+ 1152)  (+  -1736)  - (-  723)  (+  -9848) 

+ .9846 


200  + 712 
.9846 


= 926  ft. 


* In  this  problem  it  would  have  shortened  the  operation  somewhat  if  the 
meridian  of  the  survey  had  been  taken  parallel  to  the  dividing-line.  The  bear- 
ings could  have  all  been  changed  to  give  angles  from  this  meridian,  and  original 
computation  made  from  these  new  bearings. 


ZAJVn  SURVEYING. 


217 


Whence  from  eq  (9),  we  have 


MB  = 


D — EM  sin  4' 
sin  260"^ 


+ 1152  - (926) (- .0011) 

= i = II7I  ft. 

+ .9848.  ^ 


Inserting  these  values  of  the  lengths  of  the  courses  EM 
and  MB.,  we  can  compute  the  area  BCDEM.  This  is  found  to 
he  35.73  acres,  or  5.73  acres  too  much.  The  problem  now  is 
to  pass  a line  north  of  MB  and  parallel  to  it,  so  that  the  area 
included  between  the  parallel  lines  and  the  intercepted  por- 
tions of  £'/^and  BC  shall  be  5.73  acres,  or  249,710  sq.  ft.  Let 
00'  be  such  a line.  This  line  can  be  run  when  either  MO  or 
BO'  is  known.  It  is  best,  however,  to  compute  both  these 
distances,  using  one  for  a check.  To  find  these  distances. 

Let  X = perpendicular  distance  between  the  parallel  lines 
MB  and  00' . 

Let  angle  EMB^EOO’ 

and  angle  00' B =0. 

Then  we  have 


Area  MOO' B — MB  . x — ^x"^  cot  0 cot  0 

= MB  . X + (cot  0 — cot  B).  . . (i) 

Since  ^and  0 are  known  angles,  their  cotangents  are  known 
quantities  in  any  case.  So,  for  simplicity,  let 


(cot  0 — cot  B)~K\ 


2I8 


SURVEYING. 


also,  let  the  distance  MB  = D, 

and  area  MOO'B  = A. 

Then  the  equation  becomes 

Dx^\Kx^', (2) 

, 2Z?  2A 

^ ^ - K' 


D jzA  , D' 

K^\J  K' 

= - J±^4/2^Ar+Z>“; 

= i(±  ^2AK^D‘-D) (3) 

That  sign  of  the  radical  is  to  be  used  which  will  give  a 
positive  value  to  x.  The  other  sign  would  give  the  value  of 
X to  be  used  in  laying  off  the  given  area  on  the  opposite  side 
of  MB,  provided  the  sides  OM  dind  O' B were  continuous  in 
that  direction. 

Using  equation  (3)  for  the  problem  in  hand,  we  have 

e = 79°  56' ; 

0=  54°  21'; 

A — 249,710  sq.  ft. ; 

D — 1 1 71  ft. ; 

K = 0.7172  - 0.1775  = o 5397: 


LAJVI?  SURVEYING. 


2ig 


whence  ;ir  = — (±  /269, 537  + 1,371,241  - 1171) 
= 203.6  feet. 


We  can  now  find  MO  and  BO'  from 


MO  = -I—a,  and  BO'  = ; 

Sin  u sin  0 


whence  MO  — 206.8  feet,  and  BO'  = 250.6  feet. 
The  length  of  the  line  00'  is 

00'  — MB  + X (cot  0 — cot 
We  may  therefore  write  the  following 


RULE  FOR  CUTTING  OFF  A GIVEN  AREA  BY  A LINE  PASSING 
IN  A GIVEN  DIRECTION. 

Having  first  surveyed  the  tract  and  plotted  the  same,  pass 
a line  on  the  plot  in  the  required  direction  through  the  corner 
which  will  give  the  nearest  approximation  to  the  desired  area. 
Compute  the  lengths  of  the  two  unknown  courses  bounding 
this  area,  and  then  the  area  itself.  Subtract  this  from  the 
given  area,  and  the  remainder  is  the  area  which  is  to  be  cut  off 
by  a line  parallel  to  the  first  trial  line.  This  auxiliary  area  will 
always  be  a trapezoid,  whose  area,  the  length  and  bearing  of 
one  of  the  parallel  sides,  and  the  bearings  of  tlie  remaining 
sides  are  known.  The  lengths  of  these  sides  may  then  be 
computed,  one  of  the  end  lengths  laid  off,  and  the  dividing- 
line  run.  Measure  the  length  of  this  line  and  also  of  the  other 
end  line  for  checks. 


220 


SU/iVEYING. 


EXAMPLES. 

187.  Compute  the  area,  plot  the  survey,  and  determine  error  of  closure 
from  the  following  field-notes  : 


Station. 

Bearing. 

Distance. 

A 

S.  46^  E. 

20.00  ch. 

B 

S.  74^  E. 

30.95 

C 

N.  33i  E. 

18.80 

D 

N.  56  W. 

27.60 

E 

W. 

21.25 

F 

S.  51I  w. 

13.80 

( Error  of  closure  = i in  201. 


This  being  a compass-survey,  the  errors  in  latitude  and  departure  must  be 
distributed  in  proportion  to  the  lengths  of  the  courses,  regardless  of  their  bear- 
ings, or  according  to  Rule  i,  p,  192.  If  the  errors  in  the  bearings  (or  deflection 
angles)  had  been  very  small  as  compared  with  the  errors  in  measuring  the  dis- 
tances, as  is  the  case  when  the  deflection  angles  are  measured  with  a transit, 
then  Rule  3,  p.  193,  should  have  been  used. 

This  would  have  changed  the  result  by  0.08  acres,  the  result  then  being 
104.35  acres. 

188.  Find  the  area  and  error  of  closure  from  the  following  field-notes  : 


Station. 

Bearing. 

Distance. 

A 

E. 

130  rods. 

B 

00 

137 

C 

N.  81  W. 

186 

D 

S. 

54 

E 

S.  36  w. 

125 

F 

S.  45  E. 

89 

G 

N.  40  E. 

70 

LAArn  SURVEYING. 


221 


What  would  be  the  resulting  difference  in  area  from  the  use  of  Rules  i and  3 ? 

189.  In  the  example,  art.  187,  suppose  the  length  and  bearing  of  the  first 
course  were  unknown.  Let  these  be  found  as  in  Case  I.,  art.  180. 

190.  Suppose  the  length  of  course  A and  bearing  of  B are  unknown  in  same 
example.  Compute  by  Case  II. 

191.  Let  the  first  two  bearings  be  unknown.  Compute  them  by  Case  III. 

192.  Let  the  lengths  of  the  first  two  courses  be  unknown.  Find  them  by 
Case  IV. 

193.  Let  it  be  required  to  cutoff  twenty-five  acres  from  the  west  end  of 
the  tract  given  in  art.  187  by  a line  passing  through  a point  on  the  course  EC 
at  a distance  of  ten  chains  from  B.  Find  the  length  and  bearing  of  the  division- 
line, and  the  other  intersecting  point  on  the  boundary. 

194.  Let  it  be  required  to  divide  the  tract  given  in  art.  187  into  three  equal 
portions  by  north  and  south  lines.  Find  the  lengths  and  points  of  intersection 
of  such  lines  with  the  boundary-lines. 

195.  Compute  the  coordinates  of  the  corners  of  the  tract  given  in  art.  187, 
taken  with  reference  to  a point  35  chains  directly  south  of  A,  and  then  com- 
pute the  area  of  the  tract  from  these  coordinates  by  the  formula  given  in  art. 
179.  This  area  should,  of  course,  be  the  same  as  that  obtained  by  any  other 
method  where  the  same  balanced  latitudes  and  departures  are  used. 

196.  An  irregular  tract  of  land  has  a straight  line  run  through  it  and  rec- 
tangular offsets  taken  to  the  boundary.  Find  the  area  of  the  tract  from  the 
following  notes : 


Distance. 

Width. 

ch. 

ch. 

0 

2.35 

10 

8.42 

14 

12.60  , 

20 

11.38 

25 

10.75 

28 

6.15 

30.50 

0.00 

Is  it  significant  whether  or  not  this  tract  lies  on  both  sides  or  wholly  on  one 
side  of  the  base-line? 

I96ct.  Compute  the  area  of  the  tract  of  which  the  following  are  the  field- 


222 


SUR  VE  YING. 


notes.  The  rectangular  offsets  are  taken  on  both  sides  of  a straight  axial  line 
R signifying  right  and  L left. 


Distances. 

Side. 

Width  or 
Lenfjlh  of 
OlTset. 

Distances. 

Side. 

Width  or 
Lcneth  of 
Offset, 

ch. 

ch.  I 

ch. 

ch. 

O 

R 

4.23 

18 

R 

15.80 

O 

L 

0.00 

20 

L 

5-00 

5 

R 

7.16 

25 

R 

12.20 

7.50 

L 

3-45 

30 

L 

2.62 

10 

R 

12.68 

30 

R 

6.48 

10 

L 

6.00 

30 

L 

0.00 

12 

R 

10.75 

Note. — For  a valuable  paper  on  the  Judicial  Functions  of  the  Surveyor,  by 
fudge  Cooley  of  the  Michigan  Supreme  Court,  see  Appendix  A. 


CHAPTER  VIIL 


TOPOGRAPHICAL  SURVEYING  BY  THE  TRANSIT  AND 

STADIA.* 

197.  A Topographical  Survey  is  such  a one  as  gives  not 
only  the  geographical  positions  of  points  and  objects  on  the 
surface  of  the  ground,  but  also  furnishes  the  data  from  which 
the  character  of  the  surface  may  be  delineated  with  respect  to 
the  relative  elevations  or  depressions. 

198.  There  are  three  general  methods  of  making  such  a 
survey. 

First,  with  a compass  (or  transit)  and  chain,  to  determine 
geographical  position,  and  with  a level  for  obtaining  relative 
elevations. 

Second,  with'  a plane-table,  either  with  or  without  stadia- 
rods. 

Third,  with  a transit  instrument  and  stadia  rods. 

The  first  method  is  very  laborious,  slow,  and  expensive.  It 
is  therefore  not  adapted  to  large  areas.  The  second  method 
has  been  more  extensively  used  for  this  purpose  than  any 
other.  The  use  of  the  plane-table  is  fully  described  in  Chap- 
ter V.  This  method  is  giving  place,  however,  to  the  third, 
which  has  been  in  use  in  America  since  about  1864,  when  it 
was  officially  adopted  on  the  United  States  Lake  Survey. 
The  system  was  first  used  in  Italy  about  1820.  In  what  fol- 
lows, the  third  method  will  alone  be  described. 


*The  word  “stadia”  is  Italian  and  was  originally  used  to  designate  the 
rod  used  by  the  inventor  of  the  method.  It  is  now  too  firmly  established  to 
be  changed.  On  the  U.  S,  Coast  and  Geodetic  Survey  the  word  “telemeter” 
is  used  in  place  of  “stadia,”  but  this,  which  very  properly  means  distance-meas- 
urer, has  been  appropriated  for  other  appliances  used  for  measuring  at  a dis- 
tance, as  temperature,  for  example.  It  would  therefore  seem  that  “stadia” 
is  the  better  word  to  use. 


224 


SUR  VE  YING. 


199.  The  Principle  of  the  location  of  points  by  the  transit 
and  stadia,  both  horizontally  and  vertically,  is  that  of  polar 
coordinates.  That  is,  the  location  of  the  point  geographically 
is  by  obtaining  its  angular  direction  from  the  meridian  through 
the  instrument,  which  is  read  on  the  limb  of  tlie  transit,  and 
its  distance  from  the  instrument,  which  is  read  through  the 
telescope  on  the  stadia-rod  which  is  held  at  the  point.  This 
distance  is  found  by  observing  what  portion  of  the  image  of 
the  graduated  rod  is  included  between  certain  cross-hairs  in  the 
telescope.  The  farther  the  rod  is  from  the  instrument,  the 
greater  is  the  portion  of  the  rod’s  image  which  falls  between 
the  cross-wires. 

For  elevation,  the  vertical  angle  is  read  on  the  vertical  circle 
of  the  transit,  when  the  telescope  is  directed  towards  a point 
of  the  stadia-rod  as  far  from  the  ground  as  the  telescope  is 
above  the  stake  over  which  it  is  set.  The  tangent  of  this 
angle  of  elevation,  or  depression,  into  the  given  horizontal  dis- 
tance is  the  amount  by  which  the  point  is  above  or  below  the 
instrument  station. 

In  this  way,  both  the  chain  and  levelling-instrument  are  dis- 
pensed with,  and  the  slow  and  laborious  processes  of  chaining 
over  bad  ground,  and  levelling  up  and  down  hill,  are  avoided. 
The  horizontal  distances  are  obtained  as  well,  in  general,  as 
by  the  chain ; and  the  levelling  may  be  done  within  a few 
tenths  of  a foot  to  the  mile  which  is  amply  sufficient  for  topo- 
graphical purposes. 

THEORY  OF  STADIA  MEASUREMENTS. 

200.  Fundamental  Relations. — In  Fig.  61  let  LS  be  any 
lens,  or  combination  of  lenses,  used  for  the  object-glass  of  a 
telescope. 

Let  be  a portion  of  the  object  (in  this  case  the  stadia- 

rod),  and  let  be  its  image.  The  point  of  the  object  A^  has 
its  image  formed  at  A^,  and  so  with  ^2  and  B^, 


TOPOGRAPHICAL  SUR  VE  YING. 


225 


Let  the  position  of  the  image  for  parallel  rays,  or  for 
an  object  an  infinite  distance  away ; and  let  C be  the  centre  of 


the  instrument,  or  the  intersection  of  the  plumb-line,  extended, 
with  the  axis  of  the  telescope. 

Let  and  be  the  “ principal  points,”  * and  let  the 
distance  FE^  — f (focal  length), 

^ 'Z/  ] (conjugate  foci), 

— t (for  image,  intercepted  portion), 

— s (for  stadia,  intercepted  portion). 

Then,  since  A^E^  is  parallel  to  A^E^,  and  B^E^  is  parallel  to 
B^E^,  we  have 

A,B,  : A,B,  ::  IE,:  0E„ 

or,  (i) 

Also,  from  the  law  of  lenses  we  have 

* As  optics  is  generally  taught  in  the  English  text-books,  Ei  and  E^  are 
made  to  coincide  in  a point  at  or  near  the  centre  of  the  lens;  and  this  is  called 
the  “optical  centre.”  The  “principal  points”  of  the  ordinary  objective  fall 
inside  the  surfaces  of  the  lens,  but  they  never  coincide.  The  ordinary  theory 
is  sufficiently  approximate  for  the  development  of  stadia  formulae  but  it  saves 
confusion  to  make  the  conditions  rigid,  and  it  is  equally  simple. 

15 


226 


SUJ^VEV/NG. 


(2) 


On  these  two  equations  rests  the  whole  theory  of  stadia 
measurements. 

Since  the  distance  FE^  = f = focal  distance,  is  a constant 
for  any  lens  or  fixed  combination  of  lenses,  we  see  from  equa- 
tion (2)  that  if  the  object  P approaches  the  lens  the  distance 
/,  is  diminished,  and  therefore  /,  must  be  increased  ; that  is, 
the  image  recedes  farther  from  the  lens  as  the  object  ap- 
proaches it,  and  vice  versa. 

If  the  extreme  wires  in  the  reticule  of  the  telescope  be  sup- 
posed to  be  placed  at  and  in  the  figure,  then  is 

the  visual  angle  which  is  equal  to  A^E^B.^.  But  as  the  image 
changes  its  distance  from  the  objective  as  the  object  is  nearer 
to  or  farther  from  the  instrument,  so  the  reticule  is  moved 
back  and  forth,*  for  it  must  always  be  in  the  plane  of  the 
image.  Therefore  lE^  = is  a variable  quantity,  while  A^B^ 
is  constant  for  fixed  wir^s.  Therefore  the  visual  angles  at  E^ 
and  E^  are  variable. 

If  these  angles  were  constant,  the  space  intercepted  on  the 
rod,  and  the  distance  of  the  rod  from  the  objective,  would  be  in 
constant  ratio.  Since  this  is  not  true,  we  must  find  the  rela- 
tion that  does  exist  between  the  distance  Efi  and  the  space 
intercepted  on  the  rod, 

From  equation  (i)  we  have 


I s 


/.  if. 


I I I 


but  from  equation  (2) 


frf  /; 


* If  the  objective  is  moved  in  focusing  it  does  not  appreciably  affect  these 
relations. 


TOPOGRAPHICAL  SUR  VE  YING, 


22J 


S I I 


~ifr  f f: 


or 


(3) 


that  is,  the  distance  of  the  rod  from  the  objective  is  equal  to 
the  intercepted  space  in  the  rod  multiplied  by  the  constant 


objective,  and  i is  the  distance  between  extreme  wires.  If 
the  distance  between  the  extreme  wires  be  made  o.oi  of  the 
focal  length  of  the  objective,  then  the  distance  of  the  stadia- 
rod  from  the  objective  (rigidly  from  is  a hundred  times  the 
intercepted  space  on  the  rod,  plus  the  focal  length  of  the  ob- 
jective. 

Again,  if  a base  be  measured  in  front  of  the  instrument, 
with  its  initial  point  a distance  f in  front  of  the  object-glass  of 
the  telescope,  then  the  rod  may  be  held  at  any  point  on  this 
base-line,  and  its  distance  from  the  initial  point,  and  the  space 
intercepted  by  the  extreme  wires,  will  be  in  constant  ratio. 

The  lines  Aft'  and  in  Fig.  6i  show  this  relation,  for 
they  are  the  lines  defining  the  space  on  the  rod  which  is  inter- 
cepted by  the  extreme  wires  as  the  rod  moves  back  and  forth. 
Evidently  the  rod  cannot  approach  so  near  as  F , for  then  the 
image  would  be  at  an  infinite  distance  behind  the  lens.  Usu- 
ally the  extreme  position  of  reticule  does  not  correspond  to 
a position  of  rod  nearer  than  ten  to  fifteen  feet. 

It  must  be  remembered  that  any  motion  of  the  eye-piece, 
with  reference  to  the  image  and  wires,  is  only  made  to  accom- 


228 


S UR  VE  YING. 


modate  different  eyes,  and  has  no  effect  in  changing  the  rela- 
tion of  wire  interval  and  image.  The  eye-piece  is  simply  a 
magnifier  with  which  to  view  the  image  and  wires,  but  in  all 
erecting-instruments  it  also  reinverts  the  image  so  as  to  make 
it  appear  upright.  The  effect  of  the  eye-piece  has  no  place  in 
the  discussion  of  stadia  formulae. 

If  the  distance  of  the  stadia  is  to  be  reckoned  from  the 
centre  of  the  instrument,  which  it  usually  is,  and  if  this  dis- 
tance = and  the  distance  from  the  centre  of  the  instrument 
to  the  objective  {CE^  in  Fig.  6i)  =^:,  then  we  have,  from  (3), 

^ = /a  + ^ = +/+  ^ (4) 

Since  /,  /,  and  c are  constant  for  any  instrument,  we  may 
measure  f and  c directly,  and  then  find  the  value  of  i by  a 
single  observation.  Proceed  as  follows: 

1st.  Measure  the  distance  from  the  centre  of  the  instru- 
ment (intersection  of  plumb-line  with  telescope)  to  the  objec- 
tive, and  call  this  c. 

2d.  Focus  the  instrument  on  a distant  point,  preferably  the 
moon  or  a star,  and  measure  the  distance  from  the  plane  of 
the  cross-wire  to  the  objective,  and  call  this  f. 

3d.  Set  up  the  instrument,  and  measure  the  distance  /+  ^ 
forward  from  the  plumb-line,  and  set  a mark.  From  this  mark 
as  an  initial  point,  measure  off  any  convenient  base,  as  400  feet. 

4th.  Hold  the  rod  at  the  end  of  this  base,  and  measure  the 
space  intercepted  by  the  extreme  wires.  If  we  call  the  length 
of  this  base  b,  and  the  distance  intercepted  Sy  then  we  have, 
from  equation  (3), 


or 


(5) 


TOPOGRAPHICAL  SURVEYING. 


229 


Here  we  have  the  value  of  i in  terms  of  known  quantities. 
If  it  is  desirable  to  set  the  wires  at  such  a distance  apart 
s 

that-^  will  be  a given  ratio,  as^-J-g-,  then  t must  equal  o.oif.  It 

is  possible  to  set  the  wires  by  this  means  to  any  scale,  so  that 
a rod  of  given  length  may  read  any  desired  maximum  distance. 

If  it  is  desired  that  ^ should  be  determined  with  great  ac- 
curacy for  a given  instrument,  with  wires  already  set,  so  as  to 
have  a coefficient  of  reduction  for  distance,  for  readings  on  a 
rod  graduated  to  feet  and  tenths,  for  instance,  proceed  as  fol- 
lows : 

Make  two  sets  of  observations  for  distance  and  intercepted 
interval.  The  distances  should  differ  widely,  as  50  feet  and 
500  feet,  or  100  feet  and  1000  feet,  according  to  the  length  of 
rod  used.  The  shorter  distance  should  not  be  less  than  50  feet, 
and  the  longer  one  not  more  than  1000  feet  with  the  most 
favorable  conditions  of  the  atmosphere.  The  distances  are  to 
be  measured  from  the  centre  of  the  instrument.  Make  several 
careful  determinations  of  the  wire  interval  at  each  position  of 
the  rod,  and  take  the  mean  of  all  the  results  at  each  distance, 
and  call  that  the  wire  interval,  s,  for  that  distance,  d.  We  then 
have  two  equations  and  two  unknown  quantities,  these  latter 

being  and  in  the  formula,  equation  (4), 

z 


Here  the  d and  s are  observed,  and  ^ and  (/ -f"  found. 

Knowing  these,  a table  could  be  prepared  giving  values  of  d 
for  any  tabular  value  of  s for  that  instrument. 

This  applies  to  the  reading  of  distances  from  levelling-rods. 


230 


PURVEYING. 


Some  engineers  prefer,  in  this  case,  to  observe  the  wire 
interval  for  various  measured  distances,  from  the  sliortcst  to 
the  longest,  to  be  read  in  practice,  and  prepare  a table  by  inter- 
polation. If  the  observed  positions  are  sufficiently  numerous, 
this  method  should  give  identical  results  with  those  obtained 
by  the  use  of  the  formula.  The  two  methods  may  be  used  to 
check  each  other. 

From  equation  (4)  we  see  that  the  distance  of  the  rod  from 


the  centre  of  the  instrument  is  a constant  ratio 


times  the 


intercepted  space  on  the  rod,  plus  a constant  c). 

If  diagrams  or  designs  be  drawn  on  the  stadia-rod  to  the 
i i 

scale  or  so  that  \o  j.  yards  on  the  rod  would  correspond 

to  10  yards  in  distance,  and  if  the  rod  were  decorated  with 
symbols  of  this  size,  then  the  distance  of  the  rod  from  the 
instrument  could  be  read  at  once  by  noting  how  many  symbols 
were  intercepted  between  the  wires.  To  this  distance  must 
then  be  added  the  small  distance  (/+  c),  which  is  from  10  to 
16  inches  in  ordinary  field-transits.  On  all  side-readings,  taken 
only  to  locate  points  on  a map,  this  correction  need  not  be 
added,  as  one  foot  is  far  within  the  possibilites  of  plotting. 

201.  On  the  Government  Surveys  the  base  is  usually 
measured  from  the  centre  of  the  instrument^  and  its  length  is 
taken  as  about  a mean  of  those  which  the  stadia  is  intended  to 
measure,  and  the  symbols  scaled  by  this  reading.  Then,  of 
course,  the  distance  read  is  always  in  error  by  a small  amount, 
except  when  it  is  the  same  as  the  base  for  which  it  was  gradu- 
ated. For  all  shorter  distances  the  reading  is  too  small,  and 
for  all  greater  distances  the  reading  is  too  large.  Sometimes 
several  different  lengths  of  base  are  taken,  as  400,  600,  and 
800  feet,  all  from  centre  of  instrument,  and  a mean  value  of 
wire  interval  used  for  giving  the  scale  for  the  diagrams.  This 
is  practically  the  same  as  the  other,  for  in  either  case  the  scale 
is  correct  for  but  a single  distance. 


TOPOGRAPHICAL  SURVEYING. 


231 


The  correction  to  any  reading  on  a stadia  so  graduated,  in 
order  to  give  the  distance  from  the  centre  of  the  instrument,  is 


where  K = correction,  in  feet ; 

B — distance  read  on  stadia,  in  feet ; 

B'  = length  of  base,  in  feet,  for  which  the  stadia  was 


graduated. 


If  B'  = 1000  feet,  B = 100  feet,  and  c -\-f=  1.5  feet,  then 


K - 1.5  (I-  iVA)  = + 1-35  feet. 


If  B had  been  2000  feet,  then 


K = 1.5  (i  - =•-  1.5  feet. 


These  corrections  are  not  usually  applied. 

202.  Another  Method  of  determining  the  scale  for  gradu- 
ating the  rod  is  to  measure  the  base  from  the  plumb-line,  as 
above,  and  then,  from  a fixed  point  on  the  lower  part  of  the 
rod,  find  the  intervals  that  correspond  to  various  distances,  as 
100  feet,  200  feet,  300  feet,  etc.,  and  mark  these  on  the  board, 
always  keeping  the  lower  wire  on  the  fixed,  initial  point  of  the 
rod.  Then  each  lOO-foot  space  is  subdivided  into  ten  equal 
parts,  or  symbols  ; so  that,  in  reading  the  rod  afterwards,  if  the 
lower  wire  is  always  set  on  the  initial  point,  the  reading  always 
gives  the  correct  distance  from  the  centre  of  the  instrument. 

The  objection  to  this  method  is  that  the  initial  point  on 
the  rod  cannot  always  be  seen,  on  account  of  obstructions. 

203.  Adaptation  of  Formulae  to  Inclined  Sights. — The 
previous  discussion  is  applicable  to  horizontal  sights  only. 


232 


SUJ^  VE  YING. 


If  the  rod  be  held  on  the  top  of  a hill,  and  the  telescope 
pointed  towards  it,  the  reading  on  the  rod  will  give  the  linear 
distance  from  instrument  to  rod,  provided  iJic  rod  be  held  per- 
pcndicidar  to  the  line  of  sight.  As  it  would  be  inconvenient  to 
do  this,  let  the  rod  be  held  vertical  in  all  cases.  When  the 
line  of  sight  is  inclined  to  the  rod,  the  space  intercepted  is 


increased  in  the  ratio  of  i to  the  cos  of  the  angle  with  the 
horizon. 

Thus,  the  space  A' B'  (Fig.  62)  for  the  rod  perpendicular 
to  the  line  of  sight  becomes  AB  for  the  rod  vertical.  But 
A'B'  = AB  cos  v.^ 

Let  A'B'  = r',  the  reading  on  the  stadia  for  perpendicular 
position  ; and 

Let  AB  = r,  the  actual  reading  obtained  for  a vertical 
position. 

Then  r'  — r cos  v. 


But  in  equation  (4)  we  have*^,  s = r',  and  therefore  r'  c 

tr 


* This  assumes  that  A' B'  is  perpendicular  to  CB  and  CA,  which  it  is  practi- 
cally, since  the  angle  ACO'  is  so  very  small,  usually  about  15'. 


TOPOGRAPHICAL  SURVEYING. 


233 


-[“/"is  the  distance  C0'\  whereas  the  distance  on  the  horizon- 
tal, CO.,  is  generally  desired,  and  for  this  we  have 

CO  = d—  CO'  cos  V — c f)  cos  v 

— r cos’  cos  V,  (7) 

This  is  the  equation  for  reducing  all  readings  on  the  stadia 
to  the  corresponding  horizontal  distances. 

The  vertical  distance  of  O'  above  O is  equal  to  CO'  sin  F. 


But  CO'  = / + /+  ^ = r cos  2/  + /+  c, 
hence 

00'  = k = r cos  vsmv  (/+  c)  sin  v 

= sm2v c)smv.  (8) 

Equation  (8)  is  used  for  finding  the  elevation  of  the  point 
on  which  the  stadia  is  held  above  or  below  the  instrument  sta- 
tion. 

204.  Table  V.  gives  the  values  d and  h computed  from 
these  formulae  for  a stadia  reading  of  100  feet  (or  metres,  or 
yards),  with  varying  angles  up  to  30°. 

It  will  be  noted  that  the  second  term  in  the  right  member 
of  equations  (7)  and  (8)  is  always  small,  and  its  value  depends 
on  the  instrument  used.  The  values  of  this  term  are  taken 
out  separately  in  the  table ; and  three  sets  of  values  are  given 
oi(c-\-/), — viz.,  0.75  feet,  i.oo  feet,  and  1.25  feet.  If  the 
work  does  not  require  great  accuracy,  these  small  corrections 
may  be  omitted. 

The  use  of  the  table  directly  involves  a multiplication  for 


234 


SUR  VE  YING. 


every  result  obtained.  Thus,  if  the  stadia  reads  460  feet,  the 
angle  of  inclination  6°  20',  and  we  have  f c = i foot,  then 

d = 4.60  X 9878  + 0.99  = 455.4  feet, 
and  h = 4.60  X 10.96  + o.ii  = 50.53  feet. 

The  table  is  not  generally  used  for  reductions  iox  d when  the 
angle  of  elevation  is  less  than  3 to  5 degrees.  When  = 5° 
44',  this  reduction  amounts  to  just  one  per  cent.  When  an 
error  of  I in  lOO  can  be  allowed,  then  the  reduction  to  the 
horizontal  would  not  be  used  under  6°.  If  the  second  term  in 
e-^-pbe  also  neglected,  these  two  errors  tend  to  compensate  ; 
and  if  ^ 4“  / the  instrument  used  is  i foot,  and  both  these 
corrections  be  omitted,  they  do  exactly  compensate  when  the 

stadia  reading  is  100  feet,  vertical  angle  5°  44'. 


a 

<< 

200 

it 

<( 

4°  04'. 

ti 

u 

300 

<( 

<< 

3°  20'. 

u 

u 

400 

2°  52'. 

u 

it 

500 

(( 

2°  32'. 

u 

1000 

<< 

1°  46'. 

« 

n 

2000 

<< 

<< 

1°  18'. 

Therefore  the  reduction  to  the  horizontal  need  never  be 
made  when  v is  less  than  2°,  and  it  generally  may  be  neglected 
when  V is  less  than  6°. 

In  obtaining  the  difference  of  elevation,  h,  the  term  in 
c -j-  y may  be  omitted  for  all  angles  under  6°  if  errors  of  o.l 
foot  are  not  important.  For  elevations  on  the  main  line,  how- 
ever, this  term  should  always  be  included. 

In  practice,  therefore,  the  tables  are  mostly  used  to  obtain 
the  difference  of  elevation  from  the  given  stadia  reading  and 
angle  of  elevation. 


TOP  OCR  A PHICA  L SURVE  YING. 


235 


205.  Reduction  Diagram. — Since  the  use  of  these  tables 
involves  a multiplication  each  time,  and  since  a table  for  vary- 
ing distances  and  angles  would  be  very  voluminous,  it  is  prefer- 
able to  take  out  the  elevations  from  a diagram.  Such  a diagram 
has  been  prepared,  to  be  used  in  place  of  the  table.  It  is  ar- 
ranged with  both  coordinates  in  feet,  but  can  be  used  for  both 
coordinates  in  metres,  since  the  same  unit  is  used  for  both. 
It  will  only  be  neccessary  to  re-number  the  divisions,  to  adapt 
it  to  the  new  scale. 

This  diagram  has  been  prepared  with  great  care,  and  is 
arranged  to  give  distances  to  500  yards  or  metres,  or  1500  feet, 
with  elevations  to  50  feet.  For  longer  distances  or  higher 
elevations  for  a single  pointing,  the  results  may  be  obtained 
from  the  table.  Elevations  are  taken  off  from  the  diagram 
to  the  nearest  tenth  of  a foot,  with  great  readiness  ; as  the 
smallest  spaces  are  2 millimetres  square,  and  these  correspond 
to  two-tenths  of  a foot  in  elevation.  It  is  of  more  convenient 
use  than  extended  tables,  and  is  just  as  accurate  ; the  nearest 
tenth  of  a foot  being  quite  as  exact  as  one  is  warranted  in 
writing  elevations  when  obtained  in  this  manner. 

Corrections  to  the  distances  read  are  also  obtained  from 
this  diagram  for  large  vertical  angles.* 

THE  INSTRUMENTS. 

206.  The  Transit. — That  the  transit  may  be  best  adapted 
to  this  work,  there  are  certain  features  it  should  possess, 
though  all  of  them  are  by  no  means  essential.  They  will  be 
named  in  the  order  of  their  importance. 

1st.  The  horizontal  limb  should  be  graduated  from  zero  to 
360°,  preferably  in  the  direction  of  the  movement  of  the  hands 
of  a watch. 

* The  diagram  is  printed  on  heavy  lithographic  paper  20  by  24  inches,  from 
an  engraved  plate,  and  can  be  had  from  the  publishers  of  this  volume.  Price 
50  cents,  post  paid. 


236 


SURVEYING. 


2d.  The  instrument  should  have  a vertical  circle  rigidly  at- 
tached to  the  telescope  axis,  and  not  simply  an  arm  that  is 
fastened  by  a clamp-screw,  and  which  reads  on  a fixed  arc  be- 
low. So  much  depends  on  the  vertical  circle  holding  its  adjust 
ment  that  its  arrangement  should  be  the  best  possible.  Since 
the  telescope  is  not  transited,  the  vertical  circle  need  not  be 
complete. 

3d.  The  telescope  should  be  inverting,  for  two  reasons  : 
first,  in  order  to  dispense  with  two  of  the  lenses,  and  so  obtain 
a better  definition  of  image  ; and,  second,  that  the  objective 
may  have  a longer  focal  length,  thus  giving  a flatter  image  and 
a less  distorted  field. 

4th.  The  stadia  wires  should  be  fixed  instead  of  adjustable, 
as  in  the  latter  case  they  are  not  stable  enough  to  be  reliable. 

5th.  The  bubbles  on  the  plate  of  the  instrument  should  be 
rather  delicate,  so  that  a slight  change  in  level  may  become 
apparent.  They  should  also  hold  their  adjustments  well.  This 
is  very  important,  in  order  that  the  readings  of  the  vertical 
angles  may  be  reliable.  It  is  also  of  great  importance  in 
carrying  azimuth  where  the  stations  are  not  on  the  same  level. 

6th.  The  horizontal  circle  should  read  to  thirty  seconds  ; 
and  there  should  be  no  eccentricity,  so  that  one  vernier-read- 
ing shall  be  practically  as  good  as  two. 

yth.  The  instrument  (or  tripod)  should  have  an  adjustable 
centre,  for  convenience  of  setting  over  points. 

8th.  A solar  attachment  to  the  telescope  will  be  found  very 
convenient.  In  most  regions  the  azimuth  can  be  checked  up 
by  the  reading  of  the  needle,  but  in  many  places  this  is  not 
reliable. 

207.  Setting  the  Cross-wires. — The  engineer  should  al- 
ways have  at  hand  a spider’s  cocoon  of  good  wires,  and  a small 
bottle  of  thick  shellac  varnish.  If  the  dry  shellac  is  carried  it 
may  be  dissolved  in  alcohol.  If  no  such  cocoon  is  at  hand  a 
spider  may  be  caught  and  made  to  spin  a web.  The  small, 


TOPOGRAPHICAL  SUR  VE  YING. 


237 


black,  out-door  spider  makes  a good  web  for  stadia  purposes. 
A new  wire  should  be  allowed  to  dry  for  a few  minutes,  and  an 
old  one  should  be  steamed  to  make  it  more  elastic.  The 
wires  for  stadia-work  should  be  small,  round,  and  opaque. 
Some  wires  are  translucent,  and  some  are  flat  and  twisted  like 
an  auger-shank. 

Scratches  must  be  made  across  the  face  of  the  reticule 
where  the  wires  are  to  lie.  These  must  be  made  with  great 
care,  so  as  to  have  them  equally  spaced  from  the  middle  wire, 
parallel  to  each  other,  and  perpendicular  to  the  vertical  wire. 
The  distance  apart  of  the  extreme  wires  is  to  be  computed  by 
equation  (5)  for  any  desired  scale  on  the  rod. 

Take  a piece  of  web  on  the  points  of  a pair  of  dividers,  by 
wrapping  the  ends  several  times  about  the  points,  which  should 
be  separated  by  about  an  inch  ; stretch  the  wire,  by  spreading 
the  dividers,  as  much  as  it  will  bear;  and  lay  the  dividers 
across  the  reticule  in  such  a way  that  the  web  comes  in  place. 
The  dividers  must  be  supported  underneath,  so  that  the  points 
will  drop  just  a trifle  below  the  top  of  the  reticule ; otherwise 
they  would  break  the  web.  Move  the  dividers  until  the  web 
is  seen,  by  the  aid  of  a magnifying-glass  (the  eye-piece  will  do), 
to  be  in  exact  position.  Then  take  a little  shellac  on  the  end 
of  a small  stick  or  brush,  and  touch  the  reticule  over  the  web, 
being  careful  to  have  no  lateral  motion  in  the  movement. 
The  shellac  will  harden  in  a few  minutes,  when  the  dividers 
may  be  removed.  Shellac  is  not  soluble  in  water. 

208.  Graduating  the  Stadia-rod. — The  stadia-rod  is 
usually  a board  one  inch  thick,  four  or  five  inches  wide,  and 
twelve  to  fourteen  feet  long.  Sometimes  this  is  stiffened  by  a 
piece  on  the  back.  To  graduate  the  rod,  it  is  necessary  to 
know  what  space  on  the  rod  corresponds  to  a hundred  feet  (or 
yards,  or  metres)  in  distance.  Either  of  the  three  methods 
cited  on  pp.  230-1  may  be  used  for  doing  this,  but  the  first  is 
recommended.  Thus,  measure  off  c /in  front  of  the  plumb- 


238 


SUI^VILYING. 


line,  and  set  a point.  From  this  point  measure  off  any  con- 
venient base,  as  200  yards,  on  level  ground,  and  hold  the  blank 
rod  (which  has  had  at  least  two  coats  of  white  paint),  at  the  end 
of  this  base-line.  Have  a fixed  mark  or  target  on  the  upper 
part  of  the  rod,  on  which  the  upper  wire  is  set.  Have  an  assist- 
ant record  the  position  of  the  lower  wire  as  he  is  directed  by 
the  observer.  Some  sort  of  an  open  target  is  good  for  this  pur- 
pose, but  any  scheme  is  sufficient  that  will  enable  the  observer 
to  fix  the  position  of  the  extreme  wires  at  the  same  moment  with 
exactness.  This  work  should  be  done  when  there  is  no  wind, 
and  when  the  atmosphere  is  very  steady  : a calm,  cloudy  day 
is  best.  Repeat  the  operation  until  the  number  of  results,  or 
their  accordance,  shows  that  the  mean  will  give  a good  result. 
If  the  base  was  200  yards  long,  divide  this  space  into  two  equal 
parts,  then  each  of  these  parts  into  ten  smaller  parts,  and 
finally  each  small  space  into  five  equal  parts;  and  one  of 
these  last  divisions  represents  two  yards  in  distance.  Dia- 
grams are  then  to  be  constructed  on  this  scale,  in  such  a way 
that  the  number  of  symbols  can  be  readily  estimated  at  the 
greatest  distance  at  which  the  rod  is  to  be  read.  The  individ- 
ual symbols  should  be  at  least  three  inches  across ; so  that,  if 
one  of  these  is  to  represent  teit  ufiits^  as  yards  or  metres,  then 
100  units  will  cover  2^  feet,  and  a rod  14  feet  long  will  read  a 
distance  of  560  units  (yards  or  metres).  If  it  is  desired  to  read 
distances  of  a quarter  of  a mile  or  more,  the  rod  should  be 
graduated  to  read  to  yards  (or  five-foot  units,  or  metres) ; but 
if  it  is  not  to  be  used  for  distances  over  500  to  looo  feet,  it 
might  be  graduated  to  read  to  feet.  This  question  must  be 
decided  before  the  wires  are  set,  and  then  they  must  be  spaced 
accordingly. 

In  measuring  the  base,  care  should  be  taken  to  test  the 
chain  or  tape  carefully  by  some  standard. 

If  the  rod  is  to  be  graduated  to  read  to  feet,  of  course 
the  base  should  be  some  even  hundreds  of  feet,  as  600. 


TOPOGRAPHICAL  SUR  VE  YING. 


239 


In  Fig.  63  are  shown  four  designs  for  stadia-rods  which 
have  been  long  in  use,  and  are  found  to  work  well.  They  are 
intended  to  be  all  in  black  on  a white  ground.^*  It  will  be 
noticed  that  the  shortest  lines  in  these  diagrams  all  cover  a 
space  of  two  units  on  the  rod.  In  diagrams  2 and  3 the  units 
are  either  yards  or  metres,  while  in  i they  are  units  of  five 
feet  each.  In  diagram  4 the  units  are  of  two  feet  each.  The 

is  4-1' 


Fig.  63. 


successive  units  are  found  at  the  middles  and  limits  of  these 
lines  and  spaces.  Wherever  the  wire  falls,  there  should  be  a 
white  ground  on  some  part  of  the  cross-section  ; and  the  more 
white  ground  the  better,  provided  the  figures  are  distinct. 
The  black  paint  may  be  put  on  heavy,  so  that  one  coat  will  be 
sufficient. 

The  50-  and  lOO-unit  marks  should  be  distinguished  by 
special  designs.  There  should  usually  be  at  least  two  boards 
with  each  instrument,  and  sometimes  three  and  four  are  needed. 
Of  course,  these  are  all  duplicates.  After  the  unit  scale  is 
obtained,  or  the  space  on  the  rod  corresponding  to  a hundred 


* Some  engineers  prefer  red  on  the  loo-unit  figures. 


240 


SUR  VE  YING. 


units  in  distance,  these  lOO-unit  spaces  should  be  so  distributed 
as  to  be  symmetrical  with  reference  to  the  C7ids  of  the  rod.  The 
reason  of  this  will  appear  later.  Having  determined  how  many 
lOO-unit  spaces  there  will  be  on  the  rod,  fix  the  position  of  the 
two  end  lOO-unit  symbols  with  reference  to  this  symmetry,  and 
then  the  rod  is  subdivided  from  these  points. 

Special  pains  should  be  taken  to  have  the  angular  points  of 
the  diagrams  well  defined  and  in  position.  These  points  are 
on  the  lines  of  subdivision  of  the  rod. 

After  one  rod  is  subdivided,  the  others  of  that  set  may  be 
laid  alongside,  and  all  fastened  rigidly  together  ; and  then,  by 
means  of  a try-square  or  T -square,  the  remaining  rods  may  be 
marked  off. 

The  wire  interval  should  be  tested  every  few  months  by 
remeasuring  a base,  as  was  done  for  graduation,  and  reading 
the  rod  on  it,  to  see  if  this  shows  the  true  measured  distance. 
This  is  to  provide  against  a possible  change  in  the  value  of  the 
wire  interval.  If  the  wires  are  stretched  reasonably  tight  when 
they  are  put  in,  they  seldom  change,  If  they  are  too  loose, 
they  swell  in  wet  weather,  and  may  sag  some.  The  reticule 
should  be  so  firm  that  the  variable  strain  on  the  adjusting- 
screws  will  not  distort  it  appreciably. 

If  the  wire  interval  is  found  to  have  changed,  either  the 
rods  must  be  regraduated,  or  else  a correction  must  be  made 
to  all  readings  of  importance.  What  are  called  the  “ side 
shots/*  which  make  up  a large  proportion  of  the  readings 
taken,  would  not  need  to  be  corrected. 

If  the  wires  are  adjustable,  any  unit  scale  may  be  chosen 
at  pleasure,  and  the  wires  adjusted  to  this  scale.  Then,  if  the 
intervals  change,  the  matter  is  corrected  by  adjusting  the 
wires.  The  adjustable  wires  are  generally  used  to  obtain  dis- 
tances from  levelling-rods,  where  it  is  desirable  that  each  foot 
on  the  rod  shall  correspond  to  a hundred  feet  in  distance.  For 
the  ordinary  stadia-rods,  fixed  wires  are  preferable. 


TOPOGRAPHICAL  S UR  FRYING. 


241 


GENERAL  TOPOGRAPHICAL  SURVEYING. 

209.  The  Topography  of  a region  includes  not  only  the 
character  and  geographical  distribution  of  the  surface-cover- 
ing, but  also  the  exact  configuration  of  that  surface  with 
reference  to  its  elevations  and  depressions.  Thus  any  point 
is  geographically  located  when  its  position  with  reference  to 
any  chosen  point  and  a meridian  through  it  is  found,  but  to 
be  topographically  located  its  elevation  above  a chosen  level 
surface  must  also  be  known.  A topographical  survey  consists 
in  locating  by  means  of  three  coordinates  a sufficient  number 
of  points  to  enable  the  intervening  surface  to  be  known  or 
inferred  from  these.  Evidently  the  points  chosen  should  be 
such  as  would  give  the  greatest  amount  of  information.  As 
for  geographical  outline,  the  corners,  turns,  or  other  critical 
points  are  chosen,  so  for  configuration  the  points  of  changes 
in  slope,  as  the  tops  of  ridges  and  bottoms  of  ravines,  or  the 
brow  and  foot  of  a hill,  are  chosen  as  giving  the  greatest 
information. 

210.  Field-work. — Let  it  be  required  to  make  a topo- 
graphical survey  of  either  a small  tract,  a continuous  shore- 
line, or  of  a large  area,  for  the  purpose  of  making  a contour 
map. 

In  case  of  the  small  tract,  any  point  may  be  taken  as  a 
point  of  reference,  and  the  survey  referred  to  it  as  an  origin. 
In  case  of  an  extended  region,  a series  of  points  should  be 
determined  with  reference  to  each  other,  both  in  geographical 
position  and  in  elevation.  These  determined  points  should 
not  be  more  than  about  three  miles  apart.  The  points  of  ele- 
vation or  bench-marks  need  not  be  identical  with  those  fixed 
in  geographical  position.  These  last  are  best  determined  by  a 
system  of  triangulation,  and  are  called  “ triangulation  stations." 
In  the  succeeding  discussion,  the  symbol  A will  be  used  for 
triangulation  station,  and  B.M.  for  bench-mark, 

16 


242 


SUR  VE  YING, 


Firsts  a system  of  triangulation  points  is  established,  the 
angles  observed,  azimuths  and  distances  computed,  and  the 
stations  plotted  to  scale  on  the  sheet  which  is  to  contain  the 
map.  This  plotting  is  best  done,  for  small  areas,  by  comput- 
ing the  rectangular  coordinates  (latitudes  and  departures), 
and  plotting  them  from  fixed  lines  which  have  been  drawn 
upon  the  map,  accurately  dividing  it  into  squares  of  lOOO  or 
5000  units  on  a side.  They  may,  however,  be  plotted  directly 
from  the  polar  coordinates  (azimuth  and  distance)  as  given  by 
the  triangulation  reduction.  For  this  purpose,  the  sheet  on 
which  the  map  is  first  drawn,  called  the  field  sheet,  should  have 
a protractor  circle  printed  upon  it,  about  twelve  mches  hi  diam- 
eter, These  protractor  sheets  of  drawing-paper  can  be  obtained 
of  most  dealers  in  drawing-materials,  or  the  protractor  circle 
may  be  printed  to  order  on  any  given  size  or  quality  of  paper.* 
These  protractor  circles  are  very  accurate,,  and  are  graduated 
to  15'  of  arc.  Plotting  can  be  done  to  about  the  nearest  5'. 

Second,  a line  of  levels  is  run,  leaving  B.M.’s  at  convenient 
points  whose  elevation  are  computed,  all  referred  to  a com- 
mon datum.  If  the  A’s  are  not  also  B.M.’s,  then  a B.M. 
should  be  left  in  the  near  vicinity  of  each  A.  This  is  not 
essential,  however. 

Third,  the  topographical  survey  is  then  made,  and  referred 
to,  or  hung  upon,  this  skeleton  system  of  A’s  and  B.M.’s. 

The  topographical  party  should  consist  of  the  observer,  a 
recorder,  two  or  three  stadia-men,  and  as  many  axemen  as 
may  be  necessary,  generally  not  more  than  two. 

The  azimuth,  preferably  referred  to  the  true  meridian,  is 
known  for  every  line  joining  two  A’s,  as  well  as  the  length  of 
such  line. 

Set  up  the  transit  over  a A,  and  set  the  horizontal  circle 


* Messrs.  Queen  & Co.,  Philadelphia,  or  Blattner  & Adam  of  St.  Louis,  can 
furnish  such  sheets. 


TOPOGRAPHICAL  S UR  VE  YING. 


243 


(which  should  be  graduated  continuously  from  0°  to  360°  in 
the  direction  of  the  hands  of  a watch)  so  that  vernier  A will 
read  the  same  as  the  azimuth  of  the  triangulation  line  by  which 
the  instrument  is  to  be  oriented.  Clamp  the  plates  in  this 
position,  and  set  the  telescope  to  read  on  the  distant  A.  Now 
clamp  the  instrument  below,  so  as  to  fix  the  horizontal  limb, 
and  unclamp  above.  The  azimuths  of  the  triangulation  lines 
are  generally  referred  to  the  south  point  as  the  zero,  and  in 
small  systems  of  this  sort  the  forward  and  back  azimuths  are 
taken  to  be  180°  apart.  When  the  instrument  has  been  set 
and  clamped,  all  subsequent  readings  taken  at  that  station  are 
given  in  azimuth  by  the  readings  of  vernier  A on  the  horizon- 
tal limb.  For  any  pointing,  therefore,  the  reading  of  this 
vernier  gives  the  azimuth  of  the  point  referred  to  the  true 
meridian,  and  the  rod  reading  gives  the  distance  of  the  point 
from  the  instrument  station.  These  enable  the  point  to  be 
plotted  on  the  map.  To  draw  the  contour  lines,  elevations 
must  also  be  known. 

If  the  elevation  of  the  A is  known,  measure  the  height  of 
instrument  (centre  of  telescope)  above  the  A on  the  stadia,*  as 
soon  as  the  instrument  is  levelled  up  over  that  station.  Sup- 
pose this  comes  to  the  212-unit  mark.  Write  in  the  note-book, 
as  a part  of  the  general  heading  for  that  station,  “ Ht.  of  Inst. 
= 212.”  Then,  for  all  readings  from  that  station  for  eleva- 
tions, bring  the  middle  horizontal  wire  to  the  212-unit  mark 
on  the  rod,  and  read  the  vertical  angle.  From  this  inclination 
and  distance,  the  height  of  the  point  above  or  below  the 
instrument  station  is  found-  If  the  rod  be  graduated  sym- 
metrically with  reference  to  the  two  ends,  one  need  not  be 
careful  always  to  keep  the  same  end  down,  and  so  errors  from 
this  cause  are  avoided. 

* Or,  if  preferred,  a light  staff,  about  five  feet  long,  may  be  carried  with  the 
instrument  for  this  purpose,  it  being  graduated  the  same  as  the  stadia  rods  for 
this  instrument. 


244 


SUJ^  VE  YJNG. 


The  record  in  the  note-book  consists  of — 

1st.  A Description  of  the  Point,  as,  “ N.E.  cor.  of  house,” 
“intersec.  of  roads,”  “top  of  bank,”  “ C.P.”  for  “contour 
point,”  which  is  taken  only  to  assist  in  drawing  the  contours, 
“ □ i6  ” for  “ stadia  station  i6,”  etc. 

2d.  Reading  of  Ver.  A. 

3d.  Distance. 

4th.  Vert.  Angle. 

These  four  columns  are  all  that  are  used  in  the  field. 
There  should  be  two  additional  columns  on  the  left-hand  page, 
for  reductions,  viz. : 

5th.  Difference  of  elevation,  corresponding  to  the  given 
vertical  angle  and  distance,  and  which  is  taken  from  a table  or 
diagram. 

6th.  Elevatioji.  which  is  the  true  elevation  of  each  point 
referred  to  the  common  datum. 

The  right-hand  page  should  be  reserved  for  sketching. 

It  will  be  found  most  convenient  to  let  the  sketching  pro- 
ceed from  the  bottom  to  the  top  of  the  page  ; as  in  this  case 
the  recorder  can  have  his  book  properly  oriented  as  he  holds 
it  open  before  him,  and  looks  forward  along  the  line.  The 
notes  may  advance  from  top  to  bottom,  or  vice  versa,  as  de- 
sired. If  there  are  many  “ side  shots”  from  each  instrument 
station,  one  page  will  not  usually  contain  the  notes  for  more 
than  two  stations,  and  sometimes  not  even  for  one. 

The  sketch  is  simply  to  aid  the  engineer  when  he  comes  to 
plot  the  work,  and  may  often  be  omitted  altogether.  One 
soon  becomes  accustomed  to  impressing  the  characteristics  of 
a landscape  on  his  memory  so  as  to  be  able  to  interpret  his 
notes  almost  as  well  as  though  he  had  made  elaborate  sketches. 
For  beginners  the  sketches  should  be  made  with  care.  The 
observer  should  usually  make  his  own  sketches  and  plot  his 
own  work. 

After  the  instrument  is  oriented  over  a station,  and  its 


TOPOGRAPHICAL  SUR  VE  YING. 


245 


height  taken  on  the  stadia,  the  stadia-men  go  about  holding 
the  rods  at  all  points  which  are  to  be  plotted  on  the  map, 
either  in  position  or  in  elevation,  or  both.  The  choice  of  points 
depends  altogether  on  the  character  of  the  survey  ; but  since 
a single  holding  of  the  rod  gives  the  three  coordinates  of  any 
point  within  a radius  of  a quarter  of  a mile,  it  is  evident  the 
method  is  complete,  and  that  all  necessary  information  can 
thus  be  obtained.  For  very  long  sights,  the  partial  wire  inter- 
vals (intervals  between  an  extreme  and  the  middle  wire)  may 
be  read  separately  on  the  stadia,  and  in  this  way  twice  as  great 
a distance  read  as  the  rod  was  designed  for.  The  limit  of 
good  reading  is,  however,  usually  determined  by  the  state  of 
the  atmosphere,  rather  than  by  the  length  of  the  rod.  When 
the  air  is  very  tremulous,  good  readings  cannot  be  made  over 
distances  greater  than  500  feet  ; while,  when  the  atmosphere 
is  very  steady,  a half-mile  may  be  read  with  equal  facility. 

Before  the  instrument  is  removed  from  the  first  station, 
the  forward  stadia-man  selects  a suitable  site  for  the  next 
instrument  station  (generally  called  stadia  station,  and  marked 
□,  to  distinguish  it  from  a triangulation  station.  A),  and  drives 
a peg  or  hub  at  this  point.  This  peg  is  to  be  marked  in  red 
chalk,  with  its  proper  number,  and  should  have  a taller  mark- 
ing-stake driven  by  the  side  of  it.  The  peg  for  the  □ should 
be  large  enough  to  be  stable  ; for  it  must  serve  as  a reference 
point,  both  in  position  and  elevation,  during  the  period  of  the 
survey.  It  is  often  desirable  to  start  a branch  line,  or  to 
duplicate  some  portion  of  the  work,  with  one  of  these  stations 
as  the  starting-point ; and,  since  each  □ is  determined,  in 
position  and  elevation,  with  reference  to  all  the  others,  one 
can  start  a branch  line  from  one  of  these  as  readily  as  from  a 
A.  It  is  not  usually  necessary  to  put  a tack  in  the  top,  but 
the  centre  may  be  taken  as  the  point  of  reference.  The  stadia- 
man  first  holds  his  stadia  carefully  over  the  centre  of  this  □, 
with  its  edge  towards  the  instrument,  so  as  to  enable  the 


246 


SURVEYING. 


observer  to  get  a more  accurate  setting  for  azimuth.  The 
observer  could  just  as  well  bisect  the  face  of  the  rod  ; but,  if 
held  in  this  position,  the  centre  of  the  rod  may  not  be  so 
nearly  over  the  centre  of  the  peg  as  when  held  edgewise. 
This  holding  of  the  rod  edgewise  for  azimuth  checks  the  care- 
lessness of  the  stadia-man,  and  is  done  only  for  readings  on 
instrument  stations. 

At  a signal  from  the  observer,  the  stadia  is  turned  with  its 
face  to  the  instrument,  and  the  observer  reads  the  distance  and 
vertical  angle. 

It  is  advisable,  in  good  work,  to  re-orient  and  relevel  the 
instrument  just  before  reading  to  the  forward  □.  The  transit 
is  very  apt  to  get  out  of  level  after  being  used  for  some  time, 
with  more  or  less  stepping  around  it,  and  the  limb  may  have 
shifted  slightly  on  the  axis,  both  of  which  might  be  so  slight 
as  to  make  no  material  difference  for  the  side  readings,  but 
which  would  be  important  in  the  continued  line  itself.  It  is 
best,  therefore,  to  level  up  again,  and  reset  on  the  back  station, 
before  reading  to  the  forward  one.  If  it  is  inconvenient  for 
the  rear  rodman  to  go  back  to  this  station  to  give  a reading,  a 
visible  mark  should  be  left  there,  to  enable  the  observer  to 
reset  upon  it  for  azimuth,  as  it  is  not  necessary  to  read  distance 
and  vertical  angle  again. 

When  the  instrument  is  moved,  it  is  set  up  over  the  new 
station,  and  the  new  height  of  instrument  determined  and 
recorded.  The  rear  stadia-man  is  now  holding  his  rod,  edge- 
wise, on  the  station  just  left ; and  by  this  the  observer  orients 
his  instrument,  making  vernier  A read  180°  different  from  its 
previous  reading  on  this  line.  Clamping  the  plates  at  this 
reading,  the  telescope  is  turned  upon  the  rod  on  the  back  sta- 
tion, and  the  lower  plate  clamped  for  this  position.  The  circle 
is  now  oriented,  so  that,  for  a zero-reading  of  vernier  A,  the 
telescope  points  south. 

It  will  be  noted  that  the  telescope  is  never  reversed  in  this  work. 


TOPOGRAPHICAL  SURVEYING. 


247 


The  distance  and  vertical  angle  should  both  be  reread,  on 
this  back  reading,  for  a check.  If  the  vertical  circle  is  not  in 
exact  adjustment,  this  second  reading  of  the  vertical  angle  will 
show  it,  for  the  numerical  value  of  the  angle  should  be  the 
same,  with  the  opposite  sign.  If  they  are  not  the  same,  then 
the  numerical  mean  of  the  two  is  the  true  angle  of  elevation, 
and  the  difference  between  this  and  the  real  readings  is  the 
index  error  of  the  vertical  circle.  This  error  may  be  corrected 
in  the  reduction,  or  the  vernier  on  the  vertical  circle  may  be 
adjusted. 

The  second  reading  of  the  vertical  angle  on  the  stadia- 
stakes  is  thus  seen  to  furnish  a constant  check  on  the  adjust- 
ment of  the  vertical  circle,  and  should  therefore  never  be 
neglected.  If  the  circle  is  out  of  adjustment  by  a small 
amount,  as  one  minute  or  less,  in  ordinary  work  it  would  not 
be  necessary  either  to  adjust  it  or  to  correct  the  readings  on 
side-shots,  for  the  elevations  of  contour  points  are  not  required 
with  such  extreme  accuracy.  The  mean  of  the  two  readings 
on  stadia-stakes  would  still  give  the  true  difference  of  elevation 
between  them,  so  that  there  would  be  no  continued  error  in 
the  work. 

The  work  proceeds  in  this  manner  until  the  next  A is 
reached.  In  coming  to  this  station,  it  is  treated  exactly  as 
though  it  were  a newH;  and  the  forward  reading  to  it,  and 
the  back  reading  from  it,  are  identical  with  those  of  any  two 
consecutive  IZl's.  Having  thus  occupied  the  second  A,  and 
having  oriented  the  instrument  by  the  last  □.  turn  the  tele- 
scope upon  some  other  A whose  azimuth  from  this  one  is 
known.  The  reading  of  vernier  A for  this  pointing  should  be 
this  azimuth,  and  the  difference  between  this  reading  and  the 
known  azimuth  of  the  line  is  the  accumulated  error  in  azimuth 
due  to  carrying  it  over  the  stadia  line.  This  error  should  not 
exceed  five  minutes  in  the  course  of  two  or  three  miles  in  good 
work. 


248 


SUR  VE  YTNG. 


The  check  in  distance  is  to  be  found  from  plotting  the  line, 
or  from  computing  the  coordinates  of  the  single  triangulation 
line,  and  also  of  the  meandered  line,  and  comparing  the  re- 
sults. 

The  elevations  are  checked  by  computing  the  elevation  of 
the  new  A from  the  stadia  line,  and  comparing  this  with  the 
known  elevation  from  the  line  of  levels. 

In  case  the  elevations  of  the  A’s  are  not  given,  but  only 
certain  B.M.’s  in  their  vicinity,  then  the  check  can  be  made 
on  these  just  the  same.  Thus,  in  starting,  read  the  stadia  on 
the  neighboring  B.M.,  and  from  this  vertical  angle  compute  the 
elevation  of  the  A over  which  the  instrument  sets,  and  then 
proceed  as  before.  In  a similar  manner,  the  check  for  eleva- 
tion at  the  end  of  the  line  may  be  made  on  a B.M.  as  well  as 
on  the  A. 

A quick  observer  will  keep  two  or  three  stadia-men  busy 
giving  him  points;  so  that  in  flat,  open  country,  with  long 
sights,  it  may  be  advisable  to  have  three  or  even  four  stadia- 
men  for  each  instrument.  In  hilly  country  more  time  will  be 
required  in  making  the  sketches,  and  hence  fewer  stadia-men 
are  required. 

After  the  instrument  is  oriented  at  each  new  station,  the 
needle  should  be  read  as  a check.  To  make  this  needle-read- 
ing  agree  with  the  readings  of  the  verniers  on  the  horizontal 
circle  (the  north  end  with  vernier  A,  and  the  south  end  with 
vernier  B,  for  instance),  graduate  an  annular  paper  disk  the 
size  of  the  needle-circle,  and  figure  it  continuously  from  0°  to 
360°,  in  the  reverse  direction  to  that  on  the  horizontal  limb  of 
the  instrument,  and  paste  it  on  the  graduated  needle-circle  in 
such  a position  that  the  north  end  of  the  needle  reads  zero 
when  the  telescope  is  pointing  south.  If  the  variation  is  6° 
east,  this  will  bring  the  zero  of  the  paper  scale  6°  east  of  south 
on  the  needle-circle.  This  position  of  the  paper  circle  is  then 
good  within  the  region  of  this  variation  of  the  needle.  When 


TOPOGRAPHICAL  SURVEYING. 


249 


the  survey  extends  into  a region  where  the  variation  is  differ- 
ent, the  scale  will  have  to  be  reset. 

With  these  conditions,  when  the  instrument  is  oriented  for 
a zero-reading  when  the  telescope  is  south,  the  reading  of  the 
north  end  of  the  needle  will  always  agree  with  the  reading  of 
vernier  A,  and  the  south  end  with  vernier  B.  It  is  so  easy  a 
matter  to  let  the  needle  down,  and  examine  at  each  □ to  see  if 
this  be  so,  that  it  well  pays  the  trouble.  No  record  need  be 
made  of  this  reading,  as  it  is  only  used  to  check  large  errors. 

2II.  Reducing  the  Notes. — The  only  reduction  necessary 
on  the  notes  is  to  find  the  elevation  of  all  the  points  taken,  with 
reference  to  the  fixed  datum,  and  sometimes  to  correct  the 
distance  read  on  the  rod  for  inclined  sights.  The  difference  of 
elevation  between  the  El  and  any  point  read  to,  as  well  as 
the  correction  to  the  horizontal  distance,  can  be  taken  from 
Table  V.  or  from  the  diagram.  The  methods  of  using  these 
have  been  explained  (see  pp.  234-5).  After  the  differences  of 
elevation  are  taken  out,  the  final  elevations  of  the  points  are 
to  be  computed  by  adding  algebraically  the  difference  of  eleva- 
tion to  the  elevation  of  □. 

The  following  is  a sample  page  with  these  reductions; 


250 


SURVEYING. 


Gazzam,  Observer. 
Baikr,  Recorder. 
Elevation  = 24'. 94. 


April  20,  1883. 

At  □ 4.  Ht.  of  Inst.  = 87. 


Object. 

Azimuth. 
Ver.  A. 

Distance. 

Vert. 

Angle. 

Difference 

of 

Elevation. 

Eleva- 

tion 

above 

Datum. 

yds. 

□ 3 

328“  10' 

199 

— 0°  10' 

— i'.56 

— 

Bridge 

127°  40' 

70 

4-0®  32' 

+ i'-9 

26'.8 

S.E.  cor.  of  house 

142“  35' 

90 

4-0*’  15' 

+ i'-2 

26'.  I 

On  road 

180°  25' 

114 

+ 0®  7' 

+ o'.  7 

25'. 6 

Water-level,  foot  of  hill. . . . 

230°  15' 

224 

- 0®  57' 

— 10'. 9 

14'. 0 

□ 5 

128°  33'  30* 

216 

+ 0°  55' 

-f- io'.38 

— 

C.P 

190°  48' 

210 

fi®  2' 

-fii'.4 

36'.  3 

At  □ 5.  Ht.  of  Inst.  = 78.  Mean  = 

+ io'.26.  35' 

.20. 

0 4 

308°  33'  30’ 

215 

- 0®  54' 

— 10'. 13 

— 

S.W.  cor.  of  house 

43“  30' 

104 

+ 3"  3' 

-f-16'.o 

5i'-2 

Edge  of  bank 

332°  10' 

98 

+ i°57' 

-f-io'.  I 

45'.3 

S.E.  cor.  of  R.R.  station. . . 

85“  30' 

158 

+ 1°  2' 

+ 8'.5 

43'.  7 

Railroad  track 

43°  55' 

40 

+ 2°  53' 

+ 6'.o 

4i'.2 

“ “ 

79°  30' 

270 

+ 0®  9' 

-f-  2'.I 

37'- 3 

06 

79°  30' 

200 

— 0°  2' 

— o'. 36 

— 

At  0 6.  Ht.  of  Inst.  = 79.  Mean  = 

- o'.54.  34'' 

.66. 

0 5 

259°  30' 

200 

+ 0°  4' 

+ o'.  72 

— 

Cor.  of  house 

277°  55' 

II2 

+ 3°  26' 

+i9'-7 

54'-4 

Top  of  hill 

87°  25' 

198 

-f-  4°  48' 

+49'- 3 

84'. 0 

Wagon  road 

58°  15' 

186 

+ 4°  25' 

+42'.9 

77'. 6 

40°  Vl' 

216—3 

4-6°  rr' 

+73'.53 

H'-'  0 / 

213 

C.P 

41°  45' 

III 

-f  4°  41' 

-f-27'.o 

61'. 7 

0 7 

5°  25' 

194 

-|-o°  12' 

+ 2'. 04 

— 

TOPOGRAPHICAL  SURVEYING. 


251 


It  will  be  noted  that  the  reading  on  □ 5 from  □ 4 has  a 
distance  of  216  yards,  and  a vertical  angle  of  -{-0°  55';  while 
on  the  back  reading,  from  □ 5 to  □ 4 the  distance  is  215  yards, 
and  the  vertical  angle  — 0°  54'.  The  distance  was  probably 
between  215  and  216  yards,  and  the  vertical  circle  was  prob- 
ably slightly  out  of  adjustment.  The  difference  of  elevation 
is  taken  out  for  both  cases,  however,  being  respectively  10.38 
feet  and  10.13  feet.  The  mean  of  these  is  10.26  feet,  which 
stands  as  a part  of  the  general  heading  at  Q 5.  The  true 
elevation  of  0 5 is  then  found  by  adding  10.26  to  24.94, 
giving  35.20  feet,  which  is  also  set  down  as  part  of  the  general 
heading. 

The  elevations  on  the  side-readings  from  this  station  can 
now  be  taken  out.  These  side-elevations  are  only  used  for 
obtaining  the  contours,  and  hence  are  only  taken  out  to  tenths 
of  a foot.  When  the  contours  are  ten  feet  apart  or  more, 
these  side-elevations  need  only  be  taken  out  to  the  nearest 
foot.  The  elevations  of  the  stadia  stations  should,  however, 
always  be  taken  out  to  hundredths,  to  prevent  an  accumula- 
tion of  errors  in  the  line. 

The  reduction  for  distance  may  also  be  taken  from  that 
portion  of  the  diagram  arranged  for  this  purpose.  This  is 
used  the  same  as  the  other  portion ; and  the  correction  is 
found,  which  is  to  be  always  subtracted  from  the  rod-reading. 
Thus,  in  the  reading  on  □ 8 from  0 6,  we  have  a reading  of 
216  yards,  and  a vertical  angle  of  6°  33'.  The  correction  here 
is  2.16  X 1-3  = 2.8  yards,  as  found  from  the  table.  Calling  this 
3 yards  it  is  subtracted  from  the  216,  leaving  213  yards  as 
the  distance  to  be  plotted.  It  is  only  the  stadia-line  distances 
that  need  ever  be  corrected  in  this  way,  the  corrections  being 
usually  so  small  that  it  is  not  important  on  the  side-shots. 

It  will  be  noted  that  two  0’s  were  set  from  0 6.  This 
was  done  because  a branch-line  was  run  from  0 6 over  the 
bluffs.  In  order  to  make  it  unnecessary  to  occupy  0 6 again 


252 


SURVEYING. 


when  the  branch-line  came  to  be  run,  □ 8 was  set  while  0 6 
was  occupied  in  the  main-line  work.  When  the  branch-line 
came  to  be  run,  the  instrument  was  taken  directly  to  Ci]  8,  and 
oriented  on  □ 6 by  the  readings  previously  taken  from  Q 6. 

The  right-hand  page  of  the  note-book,  opposite  the  notes 
given  above,  is  occupied  with  a sketch  of  the  locality,  with  the 
□’s  marked  on,  the  general  direction  of  the  contour  lines,  the 
railroad,  stream,  houses,  etc.* 

212.  Plotting  the  Stadia  Line. — It  is  customary  to  first 
plot  the  stadia  stations  alone,  from  one  El  to  the  next,  to  find 
whether  or  not  it  checks  within  reasonable  limits.  This  part 
of  the  work  should  be  done  with  extreme  care,  so  that  if  it 
does  not  check  it  cannot  be  attributed  to  the  plotting.  In 
case  it  does  not  check  within  the  desired  limit,  then  the  line  of 
investigation  will  be  about  as  follows  until  the  error  is  found: 

1st.  Replot  the  stadia  line. 

2d.  Recompute  and  replot  the  triangulation  line. 

3d.  By  examining  the  discrepancy  on  the  plot,  try  and 
decide  whether  the  error  is  in  azimuth  or  distance,  and,  if 
possible,  where  such  error  occurred,  and  its  amount. 

4th.  Examine  the  note-book  carefully,  and  see  if  there  is  any 
evidence  of  error  there. 

5th.  If  there  is  a large  probability  that  the  error  is  of  a 
certain  character,  and  that  it  occurred  at  a certain  place,  take 
the  instrument  to  that  station,  set  it  up,  and  redetermine  the 
azimuths  or  distances  which  seem  to  be  in  error. 

6th.  If  there  is  no  high  probability  of  any  certain  errors  to 
be  examined  for  in  this  way,  then  go  back  and  run  the  line 
over,  taking  readings  on  \I\'s  only.  If  the  elevations  had  been 
found  to  check,  the  vertical  angles  may  be  omitted  on  this 
duplicate  line ; and,  on  the  other  hand,  if  the  plot  came  out  all 
right,  but  the  elevations  could  not  be  made  to  check,  then  a 
duplicate  line  must  be  run  to  determine  this  alone  ; and  in  this 

* These  notes  were  taken  from  a field-book  of  a topographical  survey  of 
Cr^:ve  Coeur  Lake  by  the  engineering  students  of  Washington  University. 


TOPOGRA  PHICAL  SUR  VE  YING. 


253 


case  the  vertical  angles  between  H’s  are  all  that  need  be  read. 
In  cases  of  this  kind,  it  will  be  found  a great  help  to  have  the 
El’s  so  well  marked  that  they  can  be  readily  found. 

With  reasonable  care  in  reading  and  in  the  handling  of  the 
instrument,  it  will  never  be  necessary  to  duplicate  a line  entire, 
for  all  readings  between  H’s  are  checked.  The  vertical  angles 
and  distances  are  checked  by  reading  them  forward  and  back 
over  every  stadia  line;  and  the  azimuth  is  checked  by  the 
needle  readings,  and  also  when  the  second  A is  reached. 

If,  in  the  progress  of  the  work,  the  readings  on  the  back  Q 
for  distance  and  vertical  angle  do  not  fairly  agree  with  these 
quantities  as  read  from  the  previous  station,  the  recorder 
should  note  the  fact : and  the  observer  should  then  re-examine 
these  readings ; and,  if  found  to  be  right,  the  first  readings, 
taken  from  the  other  station,  should  be  questioned,  and  the 
mean  not  taken  in  the  reduction. 

For  plotting  the  stadia  lines  a parallel  ruler  (moving  on 
rollers)  is  very  desirable  ; otherwise,  triangles  must  be  used. 
The  plotting  is  done  by  setting  the  parallel  ruler  or  triangle 
on  the  proper  azimuth  as  found  from  the  protractor  printed  on 
the  sheet,  moving  it  parallel  to  itself  to  the  station  from  which 
the  point  is  to  be  plotted,  and  drawing  a pencil  line  in  the  right 
direction.  Then,  with  a triangular  scale, — or,  better,  with  a 
pair  of  dividers  and  a scale  of  equal  parts, — lay  off  the  correct 
distance  on  this  line ; and  this  gives  the  point. 

If  the  instrument  was  oriented  in  the  field  for  a zero  read- 
ing for  a south  pointing,  then  the  protractor  on  the  sheet  must 
have  its  south  point  marked  zero,  and  increase  around  to  360° 
in  the  same  direction  in  which  the  limb  of  the  instrument  in- 
creases, preferably  in  the  direction  of  the  movement  of  the 
hands  of  a watch. 

213.  Check  Readings. — To  enable  the  observer  to  locate 
large  errors  in  azimuth  or  distance,  or  both,  it  is  a good  prac- 
tice to  take  azimuth  readings  to  a common  object  from  a series 
of  consecutive  stations,  if  such  be  possible.  If  the  plot  does 


254 


Sl/J^  VE  YIA^G. 


not  close,  go  back  and  plot  in  these  azimuths  ; and  if  there  has 
been  no  error  in  azimuth  or  distance  between  IZl’s,  and  no  error 
in  reading  the  azimuths  for  these  pointings,  then  all  these  lines 
will  meet  in  a common  point  on  the  plot.  If  all  but  one  in- 
termediate line  meet  at  a point,  then  the  error  probably  was 
in  reading  the  azimuth  of  this  pointing  alone.  If  several  of 
the  first  pointings  intersect  in  a point,  and  the  remaining  point- 
ings of  the  set  taken  to  this  object  intersect  in  another  point, 
then  it  is  highly  probable  that  the  error  was  in  reading  the 
azimuth  or  distance  of  the  line  connecting  these  two  .sets  of 
IZl’s ; and  the  relative  position  of  the  points  of  intersection 
will  enable  the  observer  to  decide  whether  the  error  was  in 
azimuth  or  distance,  and  about  how  much.  If,  in  this  way, 
the  error  be  located,  the  instrument  can  be  taken  to  this  point, 
and  the  readings  retaken. 

214.  Plotting  the  Side-readings. — Having  plotted  the 
stadia  line  and  made  it  check,  the  next  step  is  to  go  back  and 
plot  in  the  side-readings.  For  doing  this,  a much  more  rapid 
method  may  be  used  than  that  described  above. 

Divide  the  sheet  into  squares  by  horizontal  and  vertical 
lines  spaced  uniformly  at  from  1000  to  5000  units  apart,  ac- 
cording to  scale.  These  lines  are  to  be  used  for  orienting  the 
auxiliary  protractor,  and  also  to  test  the  paper  for  stretch  or 
shrinkage. 

The  side-readings  are  now  plotted  by  the  aid  of  a paper 
protractor,  such  as  is  shown  in  Fig.  64.  This  is  made  frorn  a 
regular  field-protractor  sheet.  The  graduated  circle  printed 
on  the  sheet  is  used  ; and  this  is  some  12  inches  in  diameter, 
and  graduated  to  15  minutes.  The  sheet  is  trimmed  down  to 
near  the  graduated  circle,  and  the  edges  divided,  as  shown  in 
the  figure,  to  any  convenient  small  scale.*  This  sheet  is  to  be 

* It  is  sometimes  desirable  to  make  the  open  space  DFE  rectangular  and 
graduate  the  sides  of  the  space  ABF  instead  of  the  outer  edges.  The  pro- 
tractor can  then  be  used  nearer  the  edge  of  the  sheet. 


TOPOGRAPHICAL  SURVEYING. 


255 


laid  upon  the  plot,  with  its  centre,  C,  coinciding  with  the  □. 
It  is  oriented  by  bringing  the  corresponding  spaces  on  opposite 
edges  to  coincide  with  any  one  of  the  spaced  lines  on  the 
plot.  This  circle  then  has  its  position  parallel  to  that  of  the 
protractor  circle  printed  on  the  sheet,  and  an  azimuth  taken 
from  the  one  will  agree  with  an  azimuth  taken  from  the 
other.  When  this  auxiliary  protractor  has  been  so  centred 
and  oriented,  let  it  be  held  in  place  by  weights.  Now  the 
part  ADEB  folds  back,  on  the  line  AB,  into  the  position  indi- 
cated by  the  dotted  lines.  The  portion  DEE  is  cut  out  en- 


tirely, so  that  when  the  flap  is  turned  back  the  space  AFB 
is  left  open.  This  space  is  to  be  large  enough  to  include  the 
longest  side-readings  when  plotted  to  scale  ; that  is,  the  radius, 
CF,  of  the  circle  to  the  scale  of  the  drawing  must  exceed  the 
longest  readings.  We  now  have  a protractor  circle  about  the 
□,  with  this  station  for  its  centre. 

Take  a triangular  scale,  select  the  side  to  be  used  in  laying 
off  the  distances,  and  paste  a piece  of  strong  paper  on  the 
lower  side  at  the  zero  point.  Make  a needle-hole  through  this 


2S6 


SURVEYING. 


paper  close  to  the  edge,  at  the  zero  of  the  scale,  h'astcn  a 
needle  through  this  hole  into  the  point  which  marks  the  exact 
position  of  the  □.  The  scale  can  now  swing  freely  around  the 
needle,  on  the  auxiliary  protractor ; and  its  zero  remains  at 
the  centre  of  the  station  from  which  the  points  are  to  be 
plotted. 

To  plot  any  point,  swing  the  scale  around  to  the  proper 
azimuth,  and  at  the  proper  distance  mark  with  the  pencil  the 
position  of  the  point.  If  this  marks  a feature  of  the  land- 
scape, it  should  be  drawn  in  at  once,  before  going  farther;  and 
if  the  elevation  of  the  point  will  be  needed  in  sketching  the 
contours,  this  should  also  be  written  in.  For  contour  points, 
the  elevation  is  all  that  is  put  down. 

In  this  manner  the  points  can  be  plotted  very  rapidly.  A 
six-inch  triangular  scale,  divided  decimally,  will  be  found  best 
for  this. 

If  there  is  very  much  of  this  work  to  be  done,  it  might  be 
found  advisable  to  have  a special  scale  constructed  for  the 


{ l<]m7T 

linn  II II  I'll 

TTpTTiiinr 

II  III  I'll' Til 

niMliri'iT 

■mn — 

1 .2 

3 

4 

5 

6 

Fig.  65, 


purpose.  Fig.  65  is  one  form  of  such  a scale  drawn  one-third 
size,  which  would  be  found  very  convenient  and  cheap.  It 
should  be  graduated  on  a bevel  edge,  and  to  such  a scale  that 
the  units  of  distance  used  on  the  rod  may  be  plotted  to  the 
scale  of  the  drawing.  The  small  needle-hole,  in  line  with  the 
graduated  edge,  should  be  only  large  enough  to  fit  the 
needle-point  used,  so  that  there  would  be  no  play.  The  rule 
then  turns  on  an  accurate  centre,  which  will  not  wear.  Such 
scales,  six  inches  long,  could  be  constructed  very  cheaply  of 
German  silver  by  any  instrument-maker. 


TOPOGRAPHICAL  SUR  VE  YING. 


257 


A special  form  of  protractor,  shown  in  Fig.  66,  has  also 
been  used  with  great  success  in  France  and  on  the  Mississippi 
River  surveys. 

''  It  is  essentially  a semicircular  protractor,  provided  with 


258 


SUR  VE  YING. 


a needle-pointed  pivot  at  its  centre,  and  having  the  straight 
edge  graduated  so  that  distances  can  be  measured  off  each 
way  from  the  pivot  ; the  angular  deflection  is  given  by  the 
graduated  circle,  reading  from  a point  marked  on  the  paper. 
The  bottom  of  the  plate  is  flush  with  the  bottom  of  the  pro- 
tractor, and  the  hole  F is  at  the  centre,  and  should  be  only 
large  enough  to  admit  a fine  needle.  The  screw  D has  a hole 
drilled  in  its  axis  to  admit  the  needle-point.  It  is  also  split, 
so  that  when  it  is  screwed  down  it  will  clamp  the  needle 
firmly.  If  the  latter  is  broken,  it  can  readily  be  replaced  by  a 
new  one.  In  addition  to  the  scale  on  the  beveled  edge,  a 
diagonal  scale  is  also  provided  as  shown.  This  instrument 
combines  all  the  requisites  for  rapid  and  accurate  plotting  of 
points  located  by  polar  co-ordinates  or  by  intersections. 

In  using  this  protractor  the  needle-point  is  placed  at,  say, 
the  first  station,  and  pressed  firmly  down.  A meridian  line  is 
then  decided  upon,  and  a point  is  marked  on  it  at  the  outer 
edge  of  the  protractor  circle.  This  will  be  the  initial  point 
from  which  the  angles  will  be  read.  As  azimuth  is  read 
from  the  south  around  by  the  west,  it  is  plain  that  the  circle, 
numbered  as  shown  and  revolved  about  the  pivot  till  the 
proper  reading  coincides  with  the  meridian  line,  will  give  the 
direction  of  the  required  point  along  the  graduated  diameter, 
while  from  the  latter  the  distance  can  be  pricked  off.  A point 
can  be  plotted  in  any  direction  without  lifting  the  protractor 
from  its  position. 

In  going  to  the  second  station  it  is  not  necessary  to  draw 
a meridian  line  through  it.  The  azimuth  between  the  first 
and  second  stakes  being  known,  if  the  pivot  be  set  at  the  lat- 
ter, and  the  protractor  revolved  so  that  the  straight  edge  coin- 
cides with  the  line  passing  through  the  two  stakes,  then  the 
point  on  the  circle  corresponding  to  the  azimuth  of  the  line 
will  be  a point  on  the  meridian  line.  This  point  being  marked 
on  the  paper  is  the  origin  for  the  angles  plotted  from  the 


TOPOGRAPHICAL  SUR  VE  YING. 


259 


second  station,  and  it  is  evident  that  they  will  bear  the  proper 
relations  to  the  points  plotted  from  the  first  station. 

Other  methods  are  employed  for  plotting  the  side  shots, 
such  as  solid  half-circle  protractors,  of  paper  or  horn,  weighted 
in  position,  with  their  centres  over  the  station.  This  is  ori- 
ented on  a meridian  drawn  through  the  point,  and  then  all  the 
points  plotted  whose  azimuth  falls  between  0°  and  180°,  when 
the  protractor  is  laid  over  on  the  other  side,  and  the  remaining 
points  plotted.  In  this  case  the  ruler  is  laid  across  the  pro- 
tractor, with  some  even  division  at  the  station.  This  method 
is  more  troublesome,  less  rapid,  and  defaces  the  drawing  more, 
than  the  other  methods  given  above.  The  plotter  should  have 
an  assistant  to  read  off  to  him  from  the  note-book.  When 
all  the  elevations  have  been  plotted,  the  contour  lines  are 
sketched  in. 

The  plotting  should  keep  pace  with  the  field-work  as  close- 
ly as  possible,  being  done  at  night  and  at  other  times  when  the 
field-work  is  prevented  or  delayed.  In  difficult  ground  the 
map  could  be  carried  into  the  field  and  the  contours  sketched 
in  on  the  ground.  At  least  the  stadia  lines  should  be  plotted 
up  and  checked  before  the  observer  leaves  the  immediate  local- 
ity. Where  the  elevations  are  checked  on  B.M.’s,  these  checks 
should  be  immediately  worked  out.  This  much,  at  least,  could 
be  done  each  evening  for  that  day’s  work. 

215.  Contour  Lines. — In  engineering  drawings  the  config- 
uration of  the  surface  is  represented  by  means  of  contour  lines. 
A contour  line  is  the  projection  upon  the  plane  of  the  paper  of 
the  intersection  of  a horizontal,  or  rather  level,  plane  with  the 
surface  of  the  ground.  These  cutting  level  planes  are  taken, 
five,  ten,  twenty,  fifty,  or  one  hundred  feet  apart  vertically, 
beginning  with  the  datum-plane,  which  is  usually  taken  below 
any  point  in  the  surface  of  the  region.  Mean  sea-level  is  the 
universal  world’s  datum  which  should  always  be  used  when 
a reasonably  accurate  connection  with  the  sea  can  be  ob- 


26o 


SURVEYING. 


tained.*  Such  contour  lines  are  shown  on  Plate  11.  The  proper 
drawing  of  these  contours  requires  some  accurate  knowledge  of 
the  surface  to  be  depicted,  aside  from  the  elevations  of  isolated 
points  plotted  on  the  map.  This  knowledge  may  consist  of  a 
vivid  mental  picture  of  the  ground,  derived  from  personal  ob- 
servation, or  it  may  be  gained  from  sketches  made  upon  the 
ground.  Even  with  this  knowledge  the  draughtsman  must 
keep  vividly  in  mind  the  true  geometrical  significance  of  the 
contour  line,  in  order  to  properly  depict  the  surface  by  this 
means.  The  ability  to  draw  the  contour  lines  accurately  on  a 
field-sheet  is  the  severest  test  of  a good  topographer.  They 
are  first  sketched  and  adjusted  in  pencil  and  then  may  be 
drawn  in  ink. 

A few  fundamental  principles  may  be  stated  that  will  assist 
the  young  engineer  in  mastering  this  art. 

1.  All  points  in  one  contour  line  have  the  same  elevation 
above  the  datum-plane. 

2.  Where  ground  is  uniformly  sloping  the  contours  must 
be  equally  spaced,  and  where  it  is  a plane  they  are  also  straight 
and  parallel. 

3.  Contour  lines  never  intersect  or  cross  each  other. 

4.  Every  contour  line  must  either  close  upon  itself  or  ex- 
tend continuously  across  the  sheet,  disappearing  at  the  limits 
of  the  drawing.  It  cannot  have  an  end  within  these  limits  (an 
apparent  exception,  though  not  really  one,  is  the  following). 

5.  No  contour  should  ever  be  drawn  directly  across  a 
stream  or  ravine.  The  contour  comes  to  the  bank,  turns  up 
stream,  and  disappears  in  the  outer  stream  line.  If  the  bed  of 
the  stream,  or  ravine,  ever  rises  above  this  plane,  then  the 
contour  crosses  it ; but  in  the  case  of  a stream  the  crossing  is 
never  actually  shown.  In  the  case  of  a ravine  the  crossing  is 
shown,  if  points  have  been  established  in  its  bed. 

6.  Where  a contour  closes  upon  itself,  the  included  area 


* See  in  Chapter  XIV. , Precise  or  Geodesic  Levelling,  p.  563. 


TOPOGRAPHICAL  SUR  VE  YING. 


261 


is  either  a hill-top  or  a depression  without  outlet.  If  the 
latter,  it  would  in  general  be  a pond  or  lake.  In  other  words, 
such  contours  enclose  either  maximum  or  minimum  points  of 
the  surface. 

7.  If  a higher  elevation  seems  to  be  surrounded  by  lower 
ones  on  the  plot,  it  is  probably  a summit ; but  if  a lower  eleva- 
tion seems  to  be  surrounded  by  higher  ones,  it  is  probably  a 
a ravine,  or  else  an  error ; otherwise  it  is  a depression  without 
outlet,  in  which  case  there  would  probably  be  a pool  of  water 
shown. 

8.  Contour  lines  cut  all  lines  of  steepest  declivity,  as  well 
as  all  ridge  and  valley  lines,  at  right  angles. 

9.  Maximum  and  minimum  ridge  and  valley  contours  must 
go  in  pairs ; that  is,  no  single  lower  contour  line  can  intervene 
between  two  higher  ones,  and  no  single  higher  contour  line  can 
intervene  between  two  lower  ones. 

10.  Vertical  sections,  or  profiles,  corresponding  to  any  line 
across  the  map,  straight  or  curved,  can  be  constructed  from  a 
contour  map,  and  conversely  a contour  map  may  be  drawn 
from  the  profiles  of  a sufficient  number  of  lines. 

11.  Each  contour  is  designated  by  its  height  above  the 
datum-plane,  as  the  fifty-foot  contour,  the  sixty-foot  contour, 
etc.  In  flat  country,  where  the  contour  lines  are  few  and  wide 
apart,  always  put  the  number  of  the  contour  on  the  higher 
side,  otherwise  it  sometimes  may  be  impossible  to  tell  on  which 
side  is  the  higher  ground. 

12.  In  taking  surface-elevations  for  determining  contour 
lines,  points  should  always  be  taken  on  the  ridge  and  valley 
lines,  and  at  as  many  intermediate  points  as  may  be  desirable. 
There  are  two  general  systems  of  selecting  these  points.  By 
one  system  points  are  chosen  approximately  in  lines  or  sec- 
tions cutting  the  contours  about  at  right  angles,  the  critical 
points  being  the  tops  and  bottoms  of  slopes  ; while  by  the 
other  system  points  are  selected  nearly  in  the  same  contour 
line, — that  is,  on  the  same  horizontal  plane, — the  critical  points 


262 


SURVEYING. 


being  the  ridge  and  valley  points,  these  being  the  points  of 
maximum  and  opposite  curvature  in  the  contour  lines  them- 
selves. By  the  second  method  one  or  two  principal  contours 
maybe  followed  continuously,  the  points  being  taken  as  nearly 
as  may  be  on  these  contour  lines.  If  such  principal  contours 
are  50  feet  apart,  then  when  these  are  accurately  drawn  on  the 
map,  any  desired  number  of  additional  contours  may  be  inter- 
polated between  the  principal  ones. 

216.  The  Final  Map. — The  field-sheets  are  drawn  as  de- 
scribed above,  in  pencil,  or  partly  in  pencil  and  partly  in  ink, 
or  wholly  in  ink,  according  to  the  use  to  be  made  of  them.  If 
they  are  simply  to  serve  as  the  embodiment  of  the  field-sur- 
vey, to  be  used  only  for  the  construction  of  the  final  maps, 
they  are  usually  left  in  pencil,  a six-H  pencil  being  used.  The 
field-sheets  are  usually  small,  about  18x24  inches.  The  final 
sheets  maybe  of  any  desired  size.  Usually  several  field-sheets 
are  put  on  one  final  sheet,  which  will  be  worked  up  wholly  in 
ink,  or  color,  the  scale  remaining  the  same.  The  work  on  the 
field-sheet  is  then  simply  transferred  to  the  final  sheet  by  the 
most  convenient  means  available.  Tracing-paper  (not  linen) 
may  be  used.  This  is  carefully  tacked  or  weighted  down  over 
the  field-sheet,  and  the  principal  features,  such  as  triangulation 
stations,  stream  and  contour  lines,  roads,  buildings,  fence  lines, 
etc.,  are  traced  in  ink.  The  tracing-paper  is  then  removed  and 
laid  upon  the  final  sheet,  orienting  it  by  making  the  triangula- 
tion stations  on  the  tracing  coincide  with  the  corresponding 
stations  on  the  final  sheet,  where  they  have  been  carefully 
plotted  from  the  triangulation  reduction.  All  the  matter  on 
the  tracing  may  now  be  transferred  to  the  paper  beneath  by 
passing  over  the  inked  lines  with  a dull  point,  bearing  down 
hard  enough  to  leave  an  impression  on  the  paper  below.  If 
preferred,  the  tracing  may  have  its  under  surface  covered  with 
plumbago  (soft  pencil-scrapings),  after  the  tracing  is  made,  and 
then  with  a very  gentle  pressure  of  the  tracing-point  will  leave 
a light  pencil  line  on  the  final  sheet.  In  either  case,  when  the 


TOPOGRAPHICAL  SUR  VE  YING. 


263 


tracing  is  removed,  these  lines  may  be  inked  in  on  the  final 
sheet. 

If  the  map  is  to  be  photo-lithographed  it  must  be  drawn 
wholly  in  black,  as  given  in  Plates  11.  and  III.  If  not,  it  is  best 
to  use  some  color  in  its  execution.  The  water-lines  may  be 
drawn  in  blue,  and  the  contours  in  brown  on  arable  land,  and  in 
black  on  barren  or  rocky  land.  In  this  way  the  character  of  the 
surface  may  be  partly  given.  Where  the  slopes  are  very  steep 
the  contour  lines  become  nearly  coincident,  but  to  further  em- 
phasize the  uneven  character  of  the  ground,  cross-hatching,  or 
hachures,  may  be  employed  on  slopes  greater  than  45°  from 
the  horizontal.  All  these  conventional  practices  are  illustrated 
on  Plate  III.,  except  the  use  of  colors,  this  map  having  been 
drawn  for  the  purpose  of  being  photo-lithographed.  Plate  II.  is 
a photo-lithograph  copy  of  a student’s  map  of  the  annual  field 
survey  of  the  engineering  students  of  Washington  University. 

217.  Topographical  Symbols  are  more  or  less  conven- 
tional, and  for  that  reason  given  forms  should  be  agreed  upon. 
The  forms  given  in  Plate  III.  were  used  on  all  the  Mississippi 
River  surveys  made  under  the  Commission,  and  are  recom- 
mended as  being  elegant  and  fairly  representative  or  natural. 
Evidently  the  rice,  cotton,  sugar,  and  wild-cane  symbols  would 
find  no  place  in  maps  of  higher  latitudes.  The  cypress-tree 
symbols  may  be  used  for  pine  to  distinguish  them  from  decid- 
uous growth,  and  the  sugar-cane  symbol  could  be  used  for 
corn  if  desired.  It  is  not  important  to  distinguish  between 
different  kinds  of  cultivated  crops,  since  these  are  apt  to  change 
from  year  to  year,  but  it  is  sometimes  desirable  to  do  so  to 
give  a more  varied  and  pleasing  appearance  to  the  map.  The 
grouping  of  the  trees  in  a large  forest  is  also  varied  simply  for 
the  appearance,  to  prevent  monotony.  Colors  are  sometimes 
used  in  place  of  pen-drawn  symbols,  but  these  are  necessarily 
so  very  conventional  as  to  require  a key  to  interpret  them,  and 
besides  it  makes  the  map  look  cheap  and  unprofessional. 

218.  Accuracy  of  the  Stadia  Method. — In  measuring  dis- 


264 


SURVEYING. 


tances  by  stadia  the  errors  made  in  reading  the  rod  are  as  apt 
to  be  plus  as  minus.  They  therefore  follow  the  law  of  com- 
pensating errors,  which  is  that  the  square  root  of  the  number 
of  errors  remains  (probably)  uncompensated.  If  the  rod  was 
properly  graduated,  therefore,  the  only  error  is  that  from  read- 
ing the  position  of  the  wires.  On  inclined  sights  the  distance 
read  on  the  rod  is  accurately  reduced  to  the  horizontal  by 
means  of  proper  tables  or  diagrams.  There  is  another  pecu- 
liarity of  this  system,  and  that  is  that  the  accuracy  depends 
very  largely  on  the  state  of  the  atmosphere.  If  this  is  clear 
and  steady  the  accuracy  attainable  for  given  lengths  of  sight 
is  much  greater  than  when  it  is  either  hazy  or  very  unsteady 
from  the  effects  of  heat.  Or,  for  a given  degree  of  accuracy, 
the  lengths  of  sight  may  be  taken  much  longer  under  favorable 
atmospheric  conditions  than  under  unfavorable  ones.  It  is 
impossible,  therefore,  to  specify  any  given  degree  of  precision 
for  given  lengths  of  sight  for  all  atmospheric  conditions.  The 
results  obtained  on  the  U.  S.  Lake  Survey  are  perhaps  a 
fair  average  for  various  conditions.  On  that  service  the  errors 
of  closure  of  141  meandered  lines  was  computed  with  a mean 
result  of  one  in  650.  The  lengths  of  sight  averaged  from 
800  to  1000  feet,  with  a maximum  length  of  about  2000  feet. 
The  official  limit  of  error  of  closure  was  one  in  300.  The 
average  length  of  the  lines  run  was  one  and  a half  miles.  If 
care  is  taken  to  shorten  up  the  sights  for  unsteady  atmo- 
sphere, and  to  reduce  all  readings  to  the  horizontal,  it  would 
not  be  difficult  to  reduce  the  error  of  closure  on  lines  aver- 
aging  from  one  to  two  miles  in  length,  to  one  in  1000  or  one 
in  1200.  Since  the  absolute  error  increases  as  the  square 
root  of  the  length  of  the  line  run,  it  is  evident  that  the  relative 
error  diminishes  as  the  length  of  line  increases.  Thus,  for  a 
single  reading  of  say  400  feet  the  error  might  possibly  be  two 
feet,  but  for  100  such  sights  the  error  probably  would  be  but 
10  X 2 feet  = 20  feet,  the  distance  now  being  40,000  feet, 
giving  an  error  of  one  in  2000. 


CHAPTER  IX. 


RAILROAD  TOPOGRAPHICAL  SURVEYING, 

WITH  THE  TRANSIT  AND  STADIA. 

219.  Objects  of  the  Survey. — Since  the  transit  and  stadia 
are  the  best  means  of  making  a general  topographical  survey, 
so  they  are  the  means  that  are  best  adapted  to  make  a railroad 
survey,  so  far  as  this  is  a topographical  survey. 

The  map  of  a railroad  survey  may  serve  two  purposes : 

Firsts  to  enable  the  engineer  to  make  a better  location  of 
the  line  than  could  be  done  in  the  field. 

Secondy  to  give  all  necessary  data  relating  to  right  of  way, 
as  the  drawing  of  deeds,  assessment  of  damages,  etc. 

In  flat  or  gently  undulating  country,  it  is  not  advisable  to 
locate  by  a map  ; but  even  here  the  map  is  quite  as  essential 
for  determining  questions  relating  to  the  right  of  way. 

In  either  case,  therefore,  a good  topographical  map  of  the 
line  is  of  prime  importance,  and  all  the  data  for  this  map  may 
be  taken  on  the  preliminary  survey.* 

Both  these  ends  may  be  served  by  the  same  map.  The 
method  of  location  by  contours  (sometimes  called  “ paper  lo- 
cation”) is  often  absolutely  necessary  in  rough  ground,  but  is 
still  more  often  judicious  in  simpler  work,  inasmuch  as  a better 
location  can  often  be  made  in  this  way. 

220.  The  Field-work. — In  this  case  there  would  be  no 
A’s  or  B.M.’s  to  check  on;  but  the  errors  in  distance  and  ele- 
vation would  be  no  more,  probably,  than  are  now  made  on 

* By  " preliminary  survey”  is  here  meant  a survey  of  a belt  of  country 
which  it  is  expected  will  embrace  the  final  line,  and  not  a mere  reconnoisance 
made  to  determine  the  feasibility  of  a line,  or  which  of  several  lines  is  the  best. 


266 


SUR  VE  YING. 


preliminary  surveys.  In  fact,  the  errors  in  distance  would  not 
be  nearly  so  great,  unless  the  chain  be  tested  frequently  for 
length,  and  the  greatest  care  taken  on  irregular  ground.  If  a 
chain  ico  feet  long  has  600  wearing-surfaces,  which  most  of 
them  have,  and  if  each  of  these  surfaces  be  supposed  to  wear 
O.Oi  inch,  which  it  will  do  in  the  course  of  a 200-  or  300-mile 
survey,  then  the  chain  has  lengthened  by  six  inches,  or  the 
error  in  distance  is  now  i in  200  from  this  cause  alone.  If  we 
add  to  this  the  uncertain  errors  that  come  from  chaining  up 
and  down  hill,  and  over  obstructed  ground,  it  is  certain  that 
the  stadia  measures  will  be  much  the  more  accurate. 

In  the  matter  of  elevations,  since  the  local  change  of  ele- 
vation is  alone  significant,  and  not  the  total  difference  of  ele- 
vation of  points  at  long  distances  apart,  the  line  of  levels 
carried  by  the  stadia  would  be  amply  sufficient  for  a prelimi- 
nary survey. 

The  following  observations  are  applicable  to  the  prelimi- 
nary survey  for  final  location,  when  it  is  expected  the  line  will 
be  included  in  the  belt  of  country  surveyed: 

1st.  All  data  should  be  taken  that  will  contribute  to  the  so- 
lution of  all  questions  of  location,  such  as  elevations  for  con- 
tour lines  ; streams  requiring  culverts,  trestles,  or  bridges,  and 
the  necessary  size  of  each,  if  possible  ; all  depressions  which 
cross  the  line,  and  will  require  a water-way,  together  with  the 
approximate  size  of  the  area  drained  ; highways  and  private 
roads  or  lanes  ; buildings  of  all  kinds,  fences,  and  hedges ; 
character  of  surface,  as  rock,  clay,  sand,  etc.  ; character  of 
vegetation,  as  cultivated,  forest,  prairie,  marsh,  etc. ; the  loca- 
tion of  any  natural  rock  that  may  be  used  for  structures  on  the 
line,  such  as  culverts  or  abutments  ; high-water  marks  if  in  a 
bottom  subject  to  overflow;  and,  in  fact,  all  information  which 
will  probably  prove  of  value  in  determining  the  location,  or  in 
making  up  a report  with  estimates  to  the  board  of  directors,  or 
in  letting  contracts  for  earthwork. 


RAILROAD  TOPOGRAPHY. 


267 


2d.  All  data  that  may  be  found  useful  in  respect  to  land 
titles  or  right  of  way,  or  that  may  relate  to  claims  for  dam- 
ages, such  as  section  corners,  boundaries,  fences,  buildings, 
streets,  roads,  lanes,  farm  roads,  cultivated  and  uncultivated 
land,  as  well  as  such  as  may  be  cultivated,  public  and  private 
grounds,  orchards,  forests,  together  with  the  value  of  the  forest 
timber,  mineral  lands,  stone  quarries,  proximity  to  villages, 
etc.  Since  the  bearings  and  position  of  all  boundary-lines  are 
of  great  importance  in  the  matter  of  right  of  way,  every  such 
boundary  should  have  at  least  two  readings  upon  it  in  the 
field  ; and  these  should  be  as  far  apart  as  possible. 

221.  The  Maps. — Before  any  plotting  is  done,  two  ques- 
tions of  importance  must  be  decided.  They  are — first., 
whether  one  set  of  maps  is  to  serve  for  both  the  location  and 
for  the  further  use  of  the  company,  or  whether  a set  of  contour 
maps,  worked  up  in  pencil,  shall  serve  for  the  location,  and 
another  set  for  the  continuous  use  of  the  company;  second, 
what  shall  be  the  scale  of  the  maps  ? These  will  be  argued 
separately. 

Whether  one  or  two  sets  of  maps  will  be  decided  on,  will  de- 
pend largely  on  the  care  that  is  exercised  with  the  locating- 
sheets.  If  these  are  carefully  worked  up  for  the  location,  and 
kept  clean,  they  can  be  utilized  for  the  final  maps.  If  they 
become  too  badly  soiled  by  field  use,  new  sheets  would  prob- 
ably be  substituted  for  the  uses  of  the  company. 

If  it  is  expected,  at  the  start,  to  have  a different  set  of 
sheets  for  the  final  maps,  then  “ protractor  sheets”  should  be 
used  for  the  location.  In  this  case,  plot  on  these  sheets  only 
such  of  the  field-notes  as  will  contribute  to  the  location ; and 
these  need  only  be  plotted  in  pencil.  When  the  location  has 
been  made,  such  features  may  be  transferred  from  the  locating- 
sheets  to  the  final  maps,  as  may  be  desired.  These  would  con- 
sist mainly  in  the  stadia  stations,  the  contours,  and  the  located 
line.  The  rest  of  the  field-notes  may  then  be  plotted  on  the 
final  sheets,  and  the  whole  worked  up  in  ink. 


268 


SURVEYING. 


If,  on  the  other  hand,  one  set  of  maps  is  to  serve  both  pur- 
poses, then  it  would,  perhaps,  be  best  to  use  plain  sheets,  as 
the  protractor  circle  would  somewhat  disfigure  the  final  maps. 
The  protractor  sheets  would,  however,  furnish  a ready  means 
of  taking  off  the  bearings  of  lines  from  the  final  charts,  which 
might  be  thought  to  compensate  for  the  slight  marring  of  the 
map’s  appearance.  If  plain  sheets  are  chosen,  then  they  should 
be  divided  into  squares  by  lines  drawn  in  ink  parallel  to  the 
sides  of  the  paper,  in  the  direction  of  the  cardinal  points  of 
the  compass.  Both  the  stadia  stations  and  the  side-readings 
may  then  be  plotted  by  means  of  the  auxiliary  protractor,  this 
being  oriented  by  the  meridian  lines  on  the  sheet.  Even  here, 
only  those  readings  would  at  first  be  plotted  that  will  contrib- 
ute to  the  location,  and  these  marked  in  pencil.  After  the 
location  has  been  decided  on,  and  the  location  notes  taken  off, 
as  described  below,  then  the  stadia  stations,  contour  lines,  the 
located  line  of  road,  and  such  other  features  as  should  be  pre- 
served on  the  final  map,  are  inked  in,  and  the  map  thoroughly 
cleaned.  The  rest  of  the  field-notes  may  now  be  plotted,  and 
the  map  finished  up. 

If  the  road  runs  through  a settled  region,  the  questions  of 
right  of  way  are  among  the  first  things  to  be  settled  ; so  that 
preliminary  maps  showing  the  relation  of  the  road  belt  to  the 
property  lines  are  essential  to  the  settlement  of  damages,  and 
to  obtaining  the  right  of  way  from  the  property-holders. 
Coincident,  therefore,  with  the  making  of  maps  to  determine 
the  location  must  come  the  construction  of  preliminary  right- 
of-way  maps  or  tracings.  On  these  latter  need  be  plotted  only 
the  boundary-lines,  fences,  more  important  buildings,  roads, 
etc.,  or  just  sufficient  to  enable  the  right-of-way  agent  to  nego- 
tiate intelligibly  with  the  property-owners.*  Neither  the  lo- 

* For  an  excellent  article  on  the  subject  of  right-of-way  maps  and  permanent 
railway-property  records,  by  Charles  Paine,  see  The  Railroad  Gazette  of  Nov. 
14,  1884.  Reprinted  in  book  form  in  “ Elements  of  Railroading." 


RAILROAD  TOPOGRAPHY. 


269 


eating  nor  the  final  map  should  be  on  a continuous  roll.  The 
roll  requires  more  room  for  storage,  is  more  apt  to  get  dusty, 
and  is  much  more  inconvenient  for  reference.  When  sheets 
are  used,  the  survey  plot  covers  a more  or  less  narrow  belt 
across  the  map.  One  of  the  edges  of  the  sheet,  either  where 
the  plot  enters  upon  it  or  disappears  from  it,  should  be  trimmed 
straight,  and  the  plot  extended  quite  to  this  edge.  This  edge 
is  then  made  to  coincide  with  one  of  the  parallel  or  meridian 
lines  of  the  next  sheet  ; so  that  when  the  line  is  plotted,  the 
sheets  may  be  tacked  down  in  such  a way  as  to  show  the  con- 
tinuous plot  of  the  survey. 

The  scale  of  the  map  will  depend  on  whether  or  not  separate 
sets  of  charts  are  to  serve  the  purposes  of  location  and  of  the 
continuous  use  of  the  company.  For  the  purpose  of  location, 
a scale  of  400  feet  to  one  inch  does  very  well ; but  for  the  final 
detail  sheets  the  scale  should  be  larger.  If  both  purposes  are 
to  be  served  by  one  set  of  maps,  then  the  scale  should  be 
about  200  feet  to  one  inch,*  with  5-  or  lo-foot  contours.  The 
sheets  should  be  about  twenty  by  twenty-four  inches. 

222.  Plotting  the  Survey. — In  case  the  map  is  plotted  on 
a protractor  sheet,  the  methods  of  plotting  will  be  identical 
with  those  for  general  topographical  work,  except  that  here 
there  will  be  no  checks,  either  for  distance,  azimuth,  or  eleva- 
tion, except  such  as  are  carried  along  or  independently  de- 
termined. For  distance,  there  is  no  check,  except  the  dupli- 
cate readings  between  instrument  stations,  unless  the  survey 
is  through  a region  which  has  already  been  surveyed.  In  this 
case  the  section  lines  may  serve  as  a check  on  the  distances. 

The  azimuth  should  be  checked  at  every  station  by  reading 
the  needle,  as  described  on  p.  248,  and  also  by  independently 
determining  the  meridian  frequently,  either  by  a solar  attach- 
ment or  by  a stellar  observation.  If  the  line  is  not  nearly 

* Some  engineers  prefer  a scale  of  100  feet  to  one  inch  for  the  final  charts  of 
the  company. 


270 


SUR  VE  YING. 


nortli  and  south,  or,  in  other  words,  if  it  is  extended  materially 
in  longitude,  then  the  azimuth  must  be  constantly  corrected 
for  convergence  of  meridians,  as  is  shown  in  Chap.  XIV. 

The  elevations  can  only  be  checked  by  the  duplicate  read- 
ings between  instrument  stations.*  All  the  greater  care 
should  be  used,  therefore,  on  readings  between  stations. 

The  first  plotting,  whether  there  are  to  be  two  sets  of  maps 
or  one,  will  consist  in  representing  on  the  sheet  only  such  data 
as  will  assist  in  deciding  on  the  location.  These  will  be  mainly 
contour  points,  streams,  important  buildings  near  the  line, 
principal  highways,  other  lines  of  railway,  villages  with  their 
streets  and  alleys  near  the  proposed  location,  the  lines  of  de- 
markation  between  cultivated  and  timbered  or  wild  land,  etc. 
From  the  plotted  elevations,  aided  by  the  sketches  in  the  note- 
book, the  contour  lines  are  drawn  in  ; if  necessary,  this  may 
be  done  on  the  ground.  This  is  sufficient  for  determining 
upon  a location. 

When  this  has  been  done,  then  the  natural  features,  the 
contour  lines,  the  stadia  stations,  and  the  located  line,  may  be 
inked  in  (or  transferred  by  means  of  tracing-paper,  in  case  the 
final  maps  are  to  be  on  separate  sheets),  and  the  remainder  of 
the  notes  plotted. 

In  drawing  the  contour  lines  in  ink,  make  those  upon  bar- 
ren or  rocky  land  in  black,  and  those  on  arable  land  in  brown. 
If  they  are  ten  feet  apart,  make  every  tenth  one  very  heavy,  and 
every  fifth  one  somewhat  heavier  than  the  others.  If  this  be  done, 
only  the  50-  and  lOO-foot  contours  need  be  numbered.  In  case  a 
map  does  not  contain  at  least  two  of  these  numbered  contours, 
then  every  contour  which  does  appear  on  the  map  should  be 
numbered,  giving  its  elevation  above  the  datum  of  the  survey. 

* It  may  be  observed  that  the  same  lack  of  sufficient  checks  on  the  distance, 
azimuth,  and  elevation  obtains  with  the  ordinary  preliminary  survey  with  tran- 
sit, level,  and  chain.  If  preferred,  all  bearings  may  be  taken  from  the  needle,  and 
then  each  alternate  station  only  need  be  occupied  by  the  instrument.  See  series  of 
articles  on  this  subject  by  the  author  in  “ The  Railroad  Gazette  ” for  Feb.  3d,  Mar. 
2d,  9th,  and  30th,  1888, 


RAILROAD  TOPOGRAPHY, 


271 


The  streams  should  be  water-lined  in  blue,  and  an  arrow 
should  tell  the  direction  of  its  flow.  The  name  should  also  be 
given  when  possible. 

All  fences  should  be  shown,  and  especial  pains  taken  to 
represent  division  fences  in  their  true  position  ; for  it  is  from 
this  map  that  the  deeds  for  the  right  of  way  are  to  be  drawn. 

Outhouses  may  be  distinguished  from  dwellings  by  diago- 
nal lines  intersecting,  and  extending  slightly  beyond  the  out- 
line. The  character  of  the  buildings  may  be^hown  by  colors, 
as  red  for  brick,  yellow  for  frame,  pale  sepia  for  stone  ; the 
outlines  always  being  in  black. 

The  stadia  stations  should  be  left  on  the  finished  sheets; 
as,  in  case  of  a disputed  boundary,  or  for  other  cause,  the  map 
may  be  replotted  if  the  positions  of  the  instrument  stations 
are  left  on  it.  The  numbers  of  the  stations  should,  of  course, 
be  appended. 

The  magnetic  bearings  of  boundary-lines  may  be  given  on 
the  map,  or  they  may  be  determined,  as  occasion  requires,  by 
means  of  the  auxiliary  protractor  and  the  true  meridian  lines 
when  the  variation  of  the  needle  is  known.  For  this  purpose, 
the  magnetic  meridian  should  be  drawn  on  each  map,  diverg- 
ing from  one  of  the  meridian  lines,  and  the  amount  of  the 
variation  marked  in  degrees  and  minutes. 

223.  Making  the  Location. — When  a preliminary  survey 
is  made,  as  above  described,  for  the  purpose  of  making  what 
is  called  a “ paper  location,”  the  location  is  first  made  on  the 
map,  and  then  staked  out  in  the  field. 

Every  railroad  line  is  a combination  of  curves,  tangents, 
and  grades ; and  it  is  the  proper  combination  of  these  which 
makes  a good  location.  If  it  be  assumed  that  the  line  is  to  be  ‘ 
included  in  the  belt  of  country  surveyed,  then  the  map  con- 
tains all  the  data  necessary  to  enable  the  engineer  to  select  the 
best  arrangements  of  curves,  tangents,  and  grades  it  is  possible 
for  him  to  obtain  on  this  ground.  This  selection  can  be  made 


272 


SUK  VE  YING. 


with  much  more  certainty  than  is  possible  on  the  ground, 
where  the  view  is  generally  obstructed,  and  where  grades  arc 
so  deceptive. 

It  is  no  part  of  this  treatise  to  discuss  the  various  problems 
that  enter  into  the  question  of  a location,  but  only  to  show 
how  to  proceed  to  make  a location  that  may  satisfy  any  given 
set  of  conditions,  by  means  of  the  contour  map. 

The  contours  themselves  will  enable  the  engineer  to  decide 
what  the  approxifnatc  grades  will  have  to  be.  Suppose  a grade 
of  0.5  foot  in  100  feet,  or  26.4  feet  to  the  mile,  has  been  fixed 
upon.  It  is  now  known  that  the  line  should  follow  the  gene- 
ral course  of  the  contours,  except  that  it  should  cross  a lo-foot 
contour  every  2000  feet.  Spread  the  dividers  to  this  distance, 
taken  to  scale,  and  mark  off  in  a rough  way  these  2000-foot 
distances  as  far  as  this  grade  is  to  extend;  and  do  the  same 
for  the  successive  grades  along  the  line.  Knowing  the  grade 
of  the  line  at  the  beginning  of  the  sheet,  the  problem  is  to  ex- 
tend this  line  over  the  sheet  so  as  to  give  the  best  location 
one  can  hope  to  get  on  this  ground  with  the  available 
means.  * 

First,  starting  from  the  initial  fixed  point  of  line  on  the 
map,  sketch  in  a line  which  will  follow  the  contours  exactly, 
crossing  them,  however,  at  such  a rate  as  to  give  the  necessary 
grade.  This  is  the  cheapest  line,  so  far  as  cut  and  fill  are  con- 
cerned. Of  course,  where  depressions  or  ridges  are  to  be 
crossed,  the  line  must  cross  over  from  a given  contour  on  one 
side  to  the  corresponding  contour  on  the  other,  and  then  fol- 
low along  the  contour  again. 

Second,  mark  out  a series  of  tangents  and  curves  which  will 
follow  this  sketched  line  as  nearly  as  it  is  possible  for  a rail- 
road to  follow  it.  This  will  not  be  the  final  location,  but  it  is 
valuable  for  study.  This  line  will  be  faulty  from  having  too 
many  and  too  sharp  curves,  and  too  little  tangent. 

Third,  draw  in  a third  line,  as  straight  as  possible,  and  with 


RAILROAD  TOPOGRAPHY. 


273 


as  low  grade  of  curves  as  possible  consistent  with  a reasonable 
amount  of  earthwork  and  a proper  distribution  of  the  same. 

For  the  purpose  of  deciding  what  degree  of  curve  is  best 
suited  to  the  ground  for  a given  deflection-angle,  it  is  well  to 
have  a series  of  paper  templets  made,  with  the  various  curves 
for  their  outer  and  inner  edges.  Of  course,  these  are  cut  with 
radii  laid  off  to  the  scale  of  the  drawing.  It  is  still  more  con- 
venient to  have  these  curves,  laid  off  to  scale,  on  a piece  of 
isinglass,  horn,  or  tracing-paper  (not  linen),  so  that  this  can  be 
laid  upon  the  map,  and  the  curve  at  once  selected  which  will 
follow  the  contours  most  economically.  Fig.  66  shows  such  a 
series  of  curves  drawn  to  a scale  of  1600  feet  to  the  inch. 


In  this  way  the  line  is  laid  out  over  the  map.  The  ques- 
tions of  greater  or  less  curvature  have  been  balanced  against  a 
less  or  greater  first  cost,  and  greater  or  less  operating  expense. 
The  question  of  shifting  it  laterally  has  also  been  examined, 
and  finally  a definite  location  fixed  upon  which  seems  to  answer 
best  to  the  case  in  hand.  When  this  is  done,  it  only  remains 
to  make  up  the  location  notes  from  which  the  line  is  to  be 
staked  out. 

18 


274 


SURVEYING. 


The  following  is  considered  a good  form  for  the  location 
notes : 

Location  Notes  for  ABC  Railroad.  From  Map  No 


Line. 

Azimuth  and 
Deflection 
Angles. 

Length. 

Station. 

Remarks. 

T 

260”  40' 

ft. 

1020 

10  20 

P.C. 

0 

P 

+ 18°  30' 

617 

lO  -f  37 

P.T. 

T 

279°  10' 

2670 

43+7 

P.C. 

4°  C.L. 

— 12°  20' 

308 

+ 15  ] 

P.T.S.  46°  30'  W. 

□ 12  320  ft. 

T 

266°  50' 

680 

52  + 95 

P.C. 

The  first  column  designates  the  tangents  and  curves,  and 
gives  the  degree  of  the  curve,  and  the  direction  of  its  curva- 
ture, whether  right  or  left.  If  it  curve  toward  the  right,  the 
azimuth  of  the  next  tangent  will  be  increased,  and  hence  its 
sign  is  plus,  and  vice  versa. 

The  second  column  gives  the  azimuths  of  the  tangents  and 
the  deflection-angles  of  the  curves.  Each  azimuth  is  seen  to 
be  the  algebraic  sum  of  the  two  preceding  angles. 

The  third  column  gives  the  lengths  of  the  tangents  as  meas- 
used  from  the  map,  and  the  lengths  of  the  curves  as  determined 
by  dividing  the  deflection-angle  by  the  degree  of  the  curve. 
Thus,  12°  20'  — 12°. 33,  and  12°. 33  4.  — 308,  which  is  the 
length  of  the  curve  in  feet.* 

The  fourth  column  gives  the  stations  and  pluses  for  the 
P.C.’s  and  the  P.T.’s.  These  quantities  are  simply  the  con- 
tinued sum  of  those  in  the  third  column. 

The  first,  second,  and  fourth  columns  now  give  all  the  infor- 

* It  is  a great  convenience  to  have  at  least  one  vernier,  in  railroad  work, 
graduated  to  read  to  hundredths  of  a degree.  The  case  here  given  is  only  one 
of  many  similar  cases;  but  the  principal  advantage  is  in  running  the  fractional 
parts  of  curves  when  the  curve  chosen  is  some  even  degree,  as  here  taken. 


RAILROAD  TOPOGRAPHY. 


275 


mation  necessary  to  stake  out  the  line.  The  stadia  is  no  longer 
to  be  used,  but  a transit  and  chain,  as  is  ordinarily  done. 

The  tangents  need  not  be  run  out  to  their  intersection  ; but 
when  the  P.C.  is  reached,  according  to  the  location  notes  taken 
from  the  map,  set  up  the  instrument,  and  stake  out  the  curve 
as  far  as  possible,  or  around  to  the  P.T.  In  either  case,  when 
the  instrument  is  to  be  moved,  make  a note  of  the  forward 
azimuth,  and  go  forward  and  orient  on  the  last  station  the 
same  as  when  moving  between  two  El’s.  If  the  instrument  be 
moved  to  the  P.T.  direct,  then,  after  orienting  back  on  the 
P.C.,  turn  off  to  the  azimuth  given  for  the  next  tangent,  and 
go  ahead.  The  tangents  could  be  run  out  to  the  intersection 
and  the  point  occupied  by  the  instrument,  for  a check,  if 
thought  desirable.  The  telescope  is  never  reversed  in  laying  out 
the  line  from  the  system  of  notes  above  given. 

With  careful  work,  the  line  ought  thus  to  be  run  out,  and 
the  curves  put  in  at  once.  We  have  supposed  there  was  no 
regular  line  cleared  out  on  the  preliminary,  so  the  necessary 
clearing  would  all  have  to  be  done  on  the  location. 

A levelling  party  follows  the  transit,  and  obtains  the  data 
for  constructing  a profile  and  for  determining  the  exact  grades. 

The  stadia  has  served  its  purpose  when  it  has  enabled  the 
engineer  to  select  the  most  favorable  position  for  the  line. 
The  transit,  chain,  and  level  must  do  the  balance.  It  is  not 
improbable  that  occasional  modifications  will  be  introduced  in 
the  field,  even  though  the  survey  and  the  location  have  been 
made  with  the  greatest  possible  care. 

224.  Another  Method  of  making  the  preliminary  survey 
from  which  to  determine  the  final  location  is  as  follows : 

Run  a transit  and  chain  line,  setting  loo-foot  stakes,  as 
nearly  on  the  line  of  the  road  as  can  be  determined  by  eye. 
Follow  this  party  by  a level  party  which  obtains  the  profile  of 
the  transit  line.  A third  party  of  one  or  more  topographers 
takes  cross-sections  at  each  loo-foot  stake  by  means  of  a 


276 


SUR  VE  YING. 


pocket-compass,  chronometer,  and  hand-level.  These  cross- 
sections  show  the  ground  on  cither  side  of  the  line  as  far  as 
desirable  by  slope  and  distance,  these  latter  being  either  meas- 
ured by  tape  or  paced.  It  is  evident  that  contour  lines  could 
be  worked  out  from  these  data,  but  these  would  not  be  needed 
if  the  distances  and  slopes  were  well  determined,  since  these 
give  a better  cross-section  than  contours  alone  could  do. 

The  objections  to  this  method  are  in  the  poor  means  it  fur- 
nishes for  accurate  determination  of  either  distances  or  slopes, 
and  the  haste  with  which  it  is  usually  done.  There  can  be  no 
question  but  that  accurate  distances  and  slopes  on  cross-sections 
100  feet  apart  would  give  fuller  data  than  even  five-foot  con- 
tours accurately  drawn.  But  to  be  accurately  determined  the 
slope  would  have  to  change  at  all  points — in  other  words,  it 
would  be  a curve.  As  to  whether  the  slopes  and  distances  as 
they  would  probably  be  taken  would  give  a better  idea  of  the 
ground  than  five-foot  contours  determined  by  the  stadia 
method,  and  the  relative  cost  of  the  two  systems,  are  matters 
of  experience.  Both  systems  are  competent  to  give  a good 
location  when  they  are  well  executed. 

Note. — The  further  study  of  railroad  surveying  falls  within  the  province  of 
the  various  railroad  field-books,  which  are  printed  in  pocket  form  and  contain 
the  necessary  tables  for  laying  out  a line  of  road.  Having  learned  the  con- 
struction and  use  of  surveying-instruments,  and  the  general  methods  of  topo- 
graphical surveying  and  levelling,  the  special  applications  to  railroad  location 
given  in  the  field-books  are  readily  mastered.  They  will  therefore  not  be 
further  considered  in  this  work. 


CHAPTER  X. 


HYDROGRAPHIC  SURVEYING. 

225.  Hydrographic  Surveying  includes  all  surveys,  for 
whatever  purpose,  which  are  made  on,  or  are  concerned  with, 
any  body  of  still  or  running  water.  Some  of  the  objects  of  such 
surveys  are  the  determination  of  depths  for  mapping  and  navi- 
gation purposes  ; the  determination  of  areas  of  cross-sections, 
the  mean  velocities  of  the  water  across  such  sections,  and  the 
slope  of  the  water  surface  ; the  location  of  buoys,  rocks,  lights, 
signals,  etc. ; the  location  of  channels,  the  directions  and  ve- 
locities of  currents,  and  the  determination  of  the  changes  in 
the  same  ; the  determination  of  the  quantity  of  sediment  car- 
ried in  suspension,  of  the  volume  of  the  scour  or  fill  on  the 
bottom,  or  of  the  material  removed  by  artificial  means,  as  by 
dredging. 

A hydrographic  survey  is  usually  connected  with  an  ex- 
tended body  of  water,  as  ocean  coasts,  harbors,  lakes,  or  riv- 
ers. The  fixed  points  of  reference  for  the  survey  are  usually 
on  shore,  but  sometimes  buoys  are  anchored  off  the  shore  and 
used  as  points  of  reference.  All  such  points  should  be  accu- 
rately located  by  triangulation  from  some  measured  base 
whose  azimuth  has  been  found.  The  buoys  will  swing  at 
their  moorings  within  small  circles,  these  being  larger  at  low 
tide  than  at  high,  but  the  errors  in  their  positions  should  never 
be  sufficient  to  cause  appreciable  error  in  the  plotted  positions 
of  the  soundings.  Where  soundings  need  to  be  located  with 
great  exactness,  buoys  could  not  be  relied  on.  The  triangula- 
tion work  for  the  location  of  the  fixed  points  of  reference  dif- 
fers in  no  sense  from  that  for  a topographical  survey.  In  fact. 


278 


SURVEYING. 


a hydrographic  survey  is  usually  connected  with  a topographical 
survey  of  the  adjacent  shores  or  banks,  the  triangulation 
scheme  serving  both  purposes.  It  is  not  uncommon,  however, 
to  make  a hydrographic  survey  for  navigation  purposes  sim- 
ply, wherein  only  the  shore-line  and  certain  very  prominent 
features  of  the  adjacent  land  are  located  and  plotted.  This  is 
the  practice  of  the  U.  S.  Hydrographic  Office  in  surveying  for- 
eign coasts  and  harbors.  In  this  case  the  work  consists  almost 
wholly  in  making  and  locating  soundings  for  a certain  limiting 
depth,  as  one  hundred  fathoms,  or  one  hundred  feet,  inward 
to  the  shore,  and  along  the  coast  as  far  as  desired.  The  length 
and  azimuth  of  a base-line  are  determined  and  the  latitude  ob- 
served by  methods  given  in  Chapter  XIV.  The  longitude  is 
found  by  observing  for  local  time,  and  comparing  it  with  the 
chronometer  time  which  has  been  brought  from  some  .station 
whose  longitude  was  known.  Whenever  telegraphic  com- 
munication can  be  obtained  with  a place  of  known  longitude, 
the  difference  between  the  local  times  of  the  two  places  is 
found  by  exchanging  chronographic  signals.  No  special  de- 
scription will  be  here  given  of  the  methods  used  in  this  part  of 
the  work,  as  they  are  all  fully  described  in  Chapter  XIV. 


THE  LOCATION  OF  SOUNDINGS. 

226.  Methods. — The  location  of  a sounding  can  be  found 
with  reference  to  visible  known  points  by  (i)  two  angles  read 
at  fixed  points  on  shore  ; (2)  by  two  angles  read  in  the  boat ; 
(3)  by  taking  the  sounding  on  a certain  range,  or  known  line, 
and  reading  one  angle  either  on  shore  or  in  the  boat  ; (4) 
by  sounding  along  a known  range,  or  line,  taking  the  soundings 
at  known  intervals  of  time,  and  rowing  at  a uniform  rate  ; (5) 
by  taking  the  soundings  at  the  intersections  of  fixed  range 
lines  ; (6)  by  means  of  cords  or  wires  stretched  between  fixed 


HYDROGRAPHIC  SURVEYING. 


279 


stations,  these  having  tags,  or  marks,  where  the  soundings  are 
to  be  taken.  These  methods  are  severally  adapted  to  differ- 
ent conditions  and  objects,  and  will  be  described  in  order. 

227.  Two  Angles  read  on  Shore. — If  two  instruments 
(transits  or  sextants)  be  placed  at  two  known  points  on  shore, 
and  the  angles  subtended  by  some  other  fixed  point,  and  the  boat 
be  read  by  both  instruments,  when  a sounding  is  taken,  the  in- 
tersection of  the  two  pointings  to  the  boat,  when  plotted  on  the 
chart  containing  the  points  of  observation  duly  plotted,  will 
be  the  plotted  position  of  the  sounding.  If  three  instruments 
are  read  from  as  many  known  stations,  then  the  three  point- 
ings to  the  boat  should  intersect  in  a point  when  plotted,  thus 
furnishing  a check  on  the  observations.  The  objections  to 
this  method  are  that  it  requires  at  least  two  observers,  and 
these  must  be  transferred  at  intervals,  as  the  work  proceeds,  in 
order  to  maintain  good  intersections,  or  in  order  to  see  the 
boat  at  all  times.  While  an  observer  is  shifting  his  posi- 
tion the  work  must  be  suspended.  If  there  are  long  lines  of 
off-shore  soundings  to  be  made  and  there  are  no  fixed  points  or 
stations  on  shore  of  sufficient  distinctness  or  prominence  to  be 
observed  by  the  sextant  from  the  boat,  then  this  method  must 
be  used.  When  the  angles  are  read  on  shore  signals  should  be 
given  preparatory  to  taking  a sounding,  and  also  when  the 
sounding  is  made.  If,  however,  the  soundings  are  taken  at 
regular  intervals  the  preparatory  signal  may  be  omitted,  and 
only  the  signal  given  when  the  sounding  is  taken.  This 
usually  consists  in  showing  a flag.  The  instrument  may  be  set 
to  read  zero  when  pointing  to  the  fixed  station.  This  reading 
need  only  be  taken  at  intervals  to  test  the  stability  of  the 
instrument. 

228.  By  Two  Angles  read  in  the  Boat  to  three  points  on 
shore  whose  relative  positions  are  known.  This  is  called  the 
“three-point”  problem.  Let  A,  C,  and  B be  the  three  shore 
points,  being  defined  by  the  two  distances  a and  b and  the  angle 


28o 


SUf!  VE  Y/iVG. 


C.  Let  the  two  angles  P and  P'  be  measured  at  the  point  P, 
The  problem  is  to  find  the  distances  AP  BP, 


{a)  Analytical  Solution. — Let  the  un- 
known angle  at  A be  ;if,  and  that  at  B be 


c 


B y.  Then  we  may  form  two  equations 
from  which  x and  y may  be  found. 
For, 


_ a sin  X b sin  y 
~ sin  P ~ sin  P' ' 


p 

Fig.  68. 


Also, 


;r+J=36o°-(P+F+C)  = ^.  ...  (2) 


From  (2),  y = R — X, 


and 


sin  y = sin  R cos  x — cos  R sin  x. 


Substitute  this  value  of  sin_y  in  (i),  reduce,  and  find 


a sin  P b sin  P cos  R 


cot  X — 


b sin  P sin  R 


■ • (3) 


When  X and  y are  found,  the  sides  AP  and  BP  are  readily 
obtained.  This  is  perhaps  the  simplest  analytical  solution  of 
the  problem. 

{U)  Geometrical  Solution. — The  following  geometrical  solu- 
tion is  of  some  interest,  though  it  is  seldom  used  : 

Let  A,  C,  and  B be  the  fixed  points  as  before,  and  Pand 
P the  observed  angles.  Having  the  points.^,  .5,  and  C plotted 


H YDROGRAPHIC  SUR  VE  YING. 


281 


in  their  true  relative  positions,  draw  from  A the  line  AD, 
making  with  AB  the  angle  P'  (CPB),  and  from  B the  line  BD, 
making  with  AB  the  angle  P (APC), 
cutting  the  former  line  in  D.  Through 
A,  D,  and  B pass  a circle,  and  through  C 
and  D draw  a line  cutting  the  circum- 
ference again  in  P.  The  point  P is  the 
plotted  position  of  the  point  of  observa- 
tion from  which  the  angles  Pand  P'  were 
measured. 

For  P must  lie  in  the  circumference 
through  ADB  by  construction,  otherwise 
would  not  be  equal  to  as  they 

are  both  measured  by  the  same  arc  AD.  The  same  holds  for 
the  angle  P'.  Also,  the  line  PD  must  pass  through  C,  other- 
wise the  angle  APC  would  be  greater  or  less  than  P,  which 
cannot  be.  The  point  Pis  therefore  on  the  line  PP,  and  also 
on  the  circumference  of  the  circle  through  ADB,  whence  it  is 
at  their  intersection. 

This  demonstration  is  valuable  as  showing  when  this 
method  of  location  fails  to  locate,  and  when  the  location  is 
poor.  For  the  nearer  the  point  D comes  to  C the  more  un- 
certain becomes  the  direction  of  the  line  CD,  and  when  D falls 
at  C — that  is,  when  P is  on  the  circumference  of  a circle  through 
A,  B,  and  C — the  solution  is  impossible,  inasmuch  as  P may 
then  be  anywhere  on  that  circumference  without 'changing  the 
angles  P and  F . This  is  also  shown  by  equation  (3),  above  ; 
for  if  A,  C,  B,  and  Pall  fall  on  one  circumference,  then  -|-j^ 
— R — 180°;  whence  cot  x—  co  X O,  which  is  indeterminate. 
For  cot  P = — CO,  and  cos  P = — i.  Also  a sin  F — b sin  P, 
both  being  equal  to  the  perpendicular  from  C on  AB.  The 
equation  then  becomes 


cot  X = 00(1  — i)  = 00  X o. 


282 


SUR  VE  YING. 


(c)  Mechanical  Solution. — If  the  three  known  stations  be 
plotted  in  position  and  the  two  observed  angles  be  carefully 
set  on  a three-armed  protractor,*  then  when  the  three  radial 
edges  coincide  with  the  three  stations,  the  centre  of  the  pro- 
tractor circle  corresponds  to  the  position  of  the  point  of  obser- 
vation. With  a good  protractor  this  method  gives  the  posi- 
tion of  the  point  as  closely  as  the  nature  of  the  observations 
themselves  would  warrant.  It  is  the  common  method  of  plot 
ting  soundings  when  two  sextant  angles  have  been  read  from 
the  sounding  boat. 

The  goniograph,  described  on  p.  113,  is  designed  to  serve 
both  for  reading  the  angles  and  for  plotting  of  point,  replacing 
therefore  both  the  sextant  and  protractor  in  this  work. 

id)  Graphical  Solution. — The  angles  may  be  laid  off  on 
tracing-paper  or  linen  by  lines  of  indeSnite  length,  and  this 
laid  on  the  plot  and  shifted  in  position  until  the  three  radial 
lines  coincide  with  the  three  stations,  when  their  intersection 
marks  the  point  of  observation.  This  is  the  most  ready 
method  of  plotting  such  observations  when  no  three-armed 
protractor  is  available. 

The  advantages  of  this  method  of  locating  soundings  are 
that  it  requires  but  one  observer,  no  time  is  lost  in  changing 
stations,  and  the  party  are  all  together,  and  hence  there  can  be 
no  misunderstandings  in  regard  to  the  work.  If  the  soundings 
are  made  in  running  water,  so  that  the  boat  cannot  be  stopped 
long  enough  to  read  two  sextant  angles,  two  sextants  are 
sometimes  used  with  one  observer,  he  setting  both  angles  and 
reading  them  afterwards  ; or  two  observers  may  be  employed 
in  the  same  boat  and  the  angles  taken  simultaneously. 

229.  By  one  Range  and  one  Angle. — The  range  may  be 
two  stations  or  poles  set  in  line  on  shore,  or  it  may  consist  of 
one  point  on  shore  and  a buoy  set  at  the  desired  position  off- 


* For  description,  with  cut,  see  p,  167. 


HYDROGRAPHIC  SUR  VE  YING. 


283 


shore.  If  buoys  are  used  they  must  be  located  by  triangulation 
from  the  shore  stations.  A triangulation  system  along  a rocky 
or  wooded  coast  may  consist  of  one  line  of  sta- 
tions on  shore  and  a corresponding  line  of  buoys. 

The  angles  are  read  only  from  the  shore  stations, 
two  angles  in  each  triangle  being  observed.  If 
the  buoys  are  well  set  and  the  work  done  in  calm 
weather,  the  results  will  be  good  enough  for  to- 
pographical or  hydrographical  purposes.  The 
stations  and  buoys  should  be  opposite  each  other, 
as  in  the  figure,  and  readings  taken  to  the  two 
adjacent  shore  stations  and  to  the  three  nearest 
buoys  from  each  shore  station.  If  the  length  of 
any  line  of  this  system  be  known,  the  rest  can  be 
found  when  the  angles  at  A,  B,  C,  and  D are 
measured.  In  such  a system  the  measured  lines 
should  recur  as  often  as  possible,  ordinary  chain- 
ing being  sufficient. 

230.  Buoys,  Buoy-flags,  and  Range-poles.— A conveni- 
ent buoy  for  this  purpose  may  be  made  of  any  light  wood, 
eighteen  inches  to  three  feet  long  in  tideless  waters,  and  long 
enough  to  maintain  an  erect  position  in  tide-waters.  It  should 
be  from  six  to  ten  inches  in  diameter  at  top,  and  taper  towards 
the  bottom.  If  the  buoy  is  not  too  long,  a hole  may  be  bored 
through  its  axis  for  the  flag-pole,  which  may  then  project  two 
or  three  feet  below  the  buoy  and  as  high  above  it  as  desired. 
The  buoy  rope  is  then  attached  to  the  bottom  end  of  the  pole 
and  made  of  such  length  as  to  maintain  the  pole  in  a vertical 
position  in  all  stages  of  the  tide.  The  anchor  may  be  any  suffi- 
ciently heavy  body,  as  a rock  or  cast-iron  disk.  If  the  buoys 
are  liable  to  become  confused  on  the  records,  different  designs 
may  be  used  in  the  flags,  as  various  combinations  of  red,  white, 
and  blue,  all  good  colors  for  this  purpose. 

The  range-poles  should  be  whitewashed  so  as  to  show  up 


Fig.  70. 


284 


SUR  VE  YING. 


against  the  background  of  the  shore.  The  ranges  are  desig- 
nated by  attaching  to  the  rear  range-poles  slats  (barrel-staves 
would  serve)  arranged  as  Roman  numerals  when  read  up  or 
down  the  pole.  If  range-poles  are  relied  on,  they  must  be  very 
carefully  located  and  plotted,  in  order  to  establish  accurately 
a long  line  of  soundings  from  a very  short  fixed  base. 

The  observed  angle  may  be  either  from  the  boat  or  from  a 
point  on  shore.  In  either  case  any  other  range-post  of  the 
series  may  be  used  either  for  the  position  of  the  observer,  if 
on  shore,  or  for  the  other  target-point  if  the  angle  is  read  from 
the  boat. 

231.  By  one  Range  and  Time-intervals. — This  is  a very 
common  and  efficient  method,  and  quite  satisfactory  where 
soundings  need  not  be  located  with  the  greatest  accuracy  and 
where  there  is  no  current.  A boat  can  be  pulled  in  still  water 
with  great  uniformity  of  speed;  and  if  the  soundings  be  taken 
at  known  intervals  with  the  ends  of  the  line  of  soundings  fixed, 
the  time-intervals  will  correspond  almost  exactly  with  the 
space-intervals.  If  the  ends  of  the  line  of  soundings  are  not 
fixed  by  buoys  or  sounding-stations  on  shore,  but  the  line  sim- 
ply fixed  by  ranges  back  from  the  water’s  edge,  the  positions  of 
the  end  soundings  may  be  fixed  by  angle-readings  and  the  bal- 
ance interpolated  from  the  time-intervals. 

232.  By  means  of  Intersecting  Ranges. — This  method 
is  only  adapted  to  the  case  where  soundings  are  to  be  repeated 
many  times  at  the  same  places.  When  the  object  of  the  sur- 
vey is  to  study  the  changes  occurring  as  to  scour  or  fill  on  the 
bottom  it  is  very  essential  that  the  successive  soundings  should 
coincide  in  position,  otherwise  discrepant  results  would  prove 
nothing.  Such  surveys  are  common  on  navigable  rivers  and 
in  harbors.  Many  systems  of  such  ranges  could  be  described, 
but  the  ingenious  engineer  will  be  able  to  devise  a system 
adapted  to  the  case  in  hand. 

233.  By  means  of  Cords  or  Wires. — In  the  case  of  a fixed 


HYDROGRAPHIC  SUR  VE  YING. 


285 


but  narrow  navigable  channel,  having  an  irregular  bottom,  or 
undergoing  improvement  by  dredging,  it  may  be  found  advis- 
able to  set  and  locate  stakes  on  opposite  sides  of  the  channel, 
to  stretch  a graduated  cord  or  wire  between  them,  and  to  locate 
the  soundings  by  this.  By  such  means  the  location  would  be 
the  most  accurate  possible. 

MAKING  THE  SOUNDINGS. 

234.  The  Lead  is  usually  made  of  lead,  and  should  be  long 
and  slender  to  diminish  the  resistance  of  the  water.  It  should 
weigh  from  five  pounds  for  shallow,  still  water,  to  twenty 
pounds  for  deep  running  water,  as  in  large  rivers.  If  depth 
only  is  required,  the  lead  may  be  a simple  cylinder  something 
like  a sash-weight  for  windows.  If  specimens  of  the  bottom 
are  to  be  brought  to  the  surface  at  each  sounding,  the  form 
shown  in  Fig.  71  may  be  used  to  advantage.  An  iron  stem, 
/,  is  made  with  a cup,  at  its  lower  end.  The 
stem  has  spurs  cut  upon  it,  or  cross-bars  attached 
to  it,  and  on  this  is  moulded  the  lead  which  gives 
the  requisite  weight.  Between  the  cup  and  the 
lead  is  a leather  cover  sliding  freely  on  the  shank 
and  fitting  tightly  to  the  upper  edges  of  the  cup. 

When  the  cup  strikes  the  bottom,  it  sinks  far 
enough  to  obtain  a specimen  of  the  same,  which 
is  then  safely  brought  to  the  surface,  the  leather 
cover  protecting  the  contents  of  the  cup  from  be- 
ing washed  out  in  raising  the  lead.  A conical  cav- 
ity in  the  lower  end  of  the  lead,  lined  with  tallow, 
is  often  used,  and  it  is  found  very  efficient  for  in- 
dicating sand  and  mud.  It  is  often  very  essential  to  know 
whether  the  bottom  is  composed  of  gravel,  coarse  or  fine,  sand, 
mud,  clay,  hard-pan,  or  rock,  and  this  knowledge  can  be  ob- 
tained with  the  cup  device  described  above. 

235.  The  Line  should  be  of  a size  suited  to  the  weight  of 


Fig.  71. 


286 


StJRVEYING. 


the  lead,  and  made  of  Italian  hemp.  It  is  prepared  for  use  by 
first  stretching  it  sufficiently  to  prevent  further  elongation  in 
use  after  it  is  graduated.  Probably  the  best  way  to  stretch  a 
line  is  to  wind  it  tightly  about  a smooth-barked  tree,  securely 
fasten  both  ends,  wet  it  thoroughly,  and  leave  it  to  dry.  Then 
rewind  as  before,  taking  up  the  slack  from  the  first  stretching, 
and  repeat  the  operation  until  the  slack  becomes  inappreciable. 
It  may  now  be  graduated  and  tagged.  Sometimes  it  is  fastened 
to  two  trees  and  stretched  by  means  of  a “ Spanish  windlass,” 
and  then  wet.  It  is  quite  possible  to  stretch  the  line  too  much, 
for  sometimes  sounding-lines  have  shortened  in  use  after  being 
stretched  by  this  method.  Soundings  at  sea  are  taken  in  fath- 
oms. On  the  U.  S.  Lake  Survey  all  depths  over  twenty- 
four  feet  (four  fathoms)  were  given  in  fathoms,  and  all 
depths  less  than  that  limit  were  given  in  feet.  On  river  and 
harbor  surveys  it  is  common  to  give  depths  in  feet.  Channel- 
soundings  on  the  Western  rivers  made  by  boatmen  are  given 
in  feet  up  to  ten  feet,  then  they  are  given  in  fathoms  and  quar- 
ters, the  calls  being  “ quarter-less-twain,”  “mark-twain,”  “quar- 
ter-twain,” “half-twain,”  “ quarter-less-three,”  “mark-three,” 
etc.,  for  depths  of  if,  2,  2f,  2f,  3,  etc.,  fathoms  respectively. 

If  the  line  is  graduated  in  feet  leather  tags  are  used  every 
five  feet,  the  intermediate  foot-marks  being  cotton  or  woollen 
strips.  The  ten-foot  tags  are  notched  with  one,  two,  three, 
etc.,  notches  for  the  10-,  20-,  30-,  etc.,  foot  points,  up  to  fifty 
feet.  The  fifty-foot  tag  may  have  a hole  in  it,  and  the  60-,  70, 
80-,  etc.,  foot-marks  have  tags  all  with  one  hole  and  with  one, 
two,  three,  etc.,  notches.  The  intermediate  five-foot  points 
have  a simple  leather  tag  unmarked.  Sometimes  the  figures 
are  branded  on  the  leather  tags,  but  notches  are  more  easily 
read.  The  zero  of  the  graduation  is  the  bottom  of  the  lead. 
The  leather  tags  are  fastened  into  the  strands  of  the  line ; the 
cloth  strips  may  be  tied  on.  The  line  should  be  frequently 
tested,  and  if  it  changes  materially  a table  of  corrections 


HYDROGRAPHIC  SURVEYING. 


287 


should  be  made  out  and  all  soundings  corrected  for  erroneous 
length  of  line. 

236.  Sounding-poles  should  be  used  when  the  depth  is 
less  than  about  fifteen  feet.  The  pole  may  be  graduated  to 
feet  simply,  or  to  feet  and  tenths,  according  to  the  accuracy 
required. 

237.  Making  Soundings  in  Running  Water. — The 

sounding-boat  should  be  of  the  “ cutter”  pattern,  with  a sort 
of  platform  in  the  bow  for  the  leadsman  to  stand  on.  If  the 
current  is  swift,  six  oarsmen  will  be  required  and  two  ob- 
servers and  one  recorder.  One  of  the  observers  may  act  as 
steersman.  If  the  depth  is  not  more  than  sixty  or  eighty  feet, 
the  soundings  are  made  without  checking  the  boat,  the  leads- 
man casting  the  lead  far  enough  forward  to  enable  it  to  reach 
bottom  by  the  time  the  line  comes  vertical.  When  the  depth 
and  the  current  are  such  as  to  make  this  impossible,  the  boat 
is  allowed  to  drift  down  with  the  current  and  soundings  taken 
at  intervals  without  drawing  up  the  lead.  The  boat  is  then 
pulled  back  up  stream  and  dropped  down  again  on  another 
line,  and  so  on. 

In  still  water  a smaller  crew  and  outfit  may  be  used,  as  the. 
boat  may  be  stopped  for  each  sounding  if  necessary. 

The  record  should  give  the  date,  names  of  observers,  general 
locality,  number  or  other  designation  of  line  sounded,  the 
time,  the  two  angles,  the  stations  sighted,  and  the  depth  for 
each  sounding,  and  the  errors  of  the  graduated  lengths  on  the 
sounding-line. 

238.  The  Water-surface  Plane  of  Reference. — In  order 
to  refer  the  bottom  elevations  to  the  general  datum-pHane  of 
the  survey,  it  is  necessary  to  know  the  elevation  of  the  water-, 
surface  at  all  times  when  soundings  are  taken.  In  tidal  waters 
the  elevation  of  “ mean  tide”  is  the  plane  of  reference  for  both 
the  topographical  and  hydrographical  surveys,  and  then  the- 
state  of  the  tide  must  be  known  with  reference  to  mean;tide.. 


288 


SUR  VE  YJNG. 


This  is  found  from  the  hourly  readings  of  a tide-gauge  (pro- 
vided it  is  not  automatic),  the  elevation  of  the  zero  of  which, 
with  reference  to  mean  tide-water,  has  been  determined.  All 
soundings  must  then  be  reduced  to  what  they  would  have  been 
if  made  at  mean  tide  before  they  are  plotted. 

If  the  soundings  are  made  in  lakes,  the  datum  is  usually 
the  lowest  water-stage  on  record  ; and  here  also  gauge-readings 
are  necessary,  as  the  stage  of  the  water  in  the  lake  varies  from 
year  to  year.  In  this  case  the  gauge  need  only  be  read  twice 
a day. 

In  rivers  of  variable  stage  the  datum  is  either  referred  to 
mean  or  low-water  stage,  or  else  to  the  general  datum  of  the 
map.  If  the  stage  is  changing  rapidly  the  gauge  should  be 
read  hourly  when  soundings  are  taken,  otherwise  daily  readings 
are  sufficient.  If  the  soundings  are  to  be  referred  to  the 
general  datum  of  the  map,  then  the  slope  of  the  stream  must 
be  taken  into  account.  If  they  are  referred  to  a particular 
stage  of  water  in  the  river,  then  the  slope  does  not  enter  as  a 
correction,  as  the  slope  is  assumed  to  be  the  same  at  all  stages, 
although  this  is  not  strictly  true. 

239.  Lines  of  Equal  Depth  correspond  to  contour  lines 
in  topographical  surveys ; but  to  draw  lines  of  equal  depth 
with  certainty  the  elevations  of  many  more  points  are  neces- 
sary than  are  needed  for  drawing  contour  lines,  because  the 
bottom  cannot  be  viewed  directly,  while  the  ground  can  be. 
Where  the  ground  is  seen  to  be  nearly  level  no  elevations 
need  be  taken,  while  for  a similar  region  of  bottom  a great 
many  soundings  would  be  required  to  prove  that  it  was  not 
irregular. 

240.  Soundings  on  Fixed  Cross-sections  in  Rivers. — 

Where  the  same  section  is  to  be  sounded  a great  many  times, 
and  especially  when  it  is  desirable  to  obtain  the  successive 
soundings  at  about  the  same  points,  it  is  best  to  fix  range- 
posts  on  the  line  of  the  section  (on  both  sides  if  it  be  a river) 


H YDROGRAPHIC  SUR  VE  YING. 


289 


and  then  fix  one  or  more  series  of  intersecting  ranges  at  points 
some  distance  above  or  below  the  section  on  one  or  both  sides 
of  the  river.  The  soundings  can  then  be  made  at  the  same 
points  continuously  without  having  to  observe  any  angles  at 
all.  Such  a system  of  ranges  is  shown  in  Fig.  72.  AA'  and 
BB'  are  range-poles  on  the  section  line.  O and  0'  are  tall 
white  posts  set  at  convenient  points  on  opposite  sides  of  the 
river,  either  above  or  below  the  section.  I.,  IL,  III.,  etc.,  are 
shorter  posts  set  near  the  bank  in  such  positions  that  the  in- 
tersection of  the  lines  O-l.,  6>-II.,  etc.,  with  the  section  range 


BB'  will  locate  the  soundings  at  i,  2,  etc.,  on  this  section  line. 
The  posts  in  the  banks  should  be  marked  by  strips  nailed  upon 
them  so  as  to  make  the  Roman  numerals  as  given  in  the  figure. 
Such  a system  of  ranges  as  the  above  is  useful  also  for  fixing 
points  on  a section-line,  for  setting  out  floats,  or  for  running 
current-meters  for  the  determination  of  river  discharge. 

241.  Soundings  for  the  Study  of  Sand-waves. — In  all 
cases  where  streams  flow  in  sandy  beds,  the  bottom  consists 
of  a series  of  wave-like  elevations  extending  across  the  chan- 
nel. These  are  very  gently  sloping  on  the  up-stream  side 
19 


290 


SURVEYING. 


and  quite  abrupt  on  the  lower  side.  They  are  called  sand- 
waves,  or  sand-reefs.  They  are  constantly  moving  down- 
stream from  the  slow  removals  from  the  upper  side  and  accre- 
tions on  the  abrupt  lower  face.  They  have  been  observed  as 
high  as  ten  feet  on  the  Mississippi  River,  and  with  a rate  of 
motion  as  great  as  thirty  feet  per  day.  In  order  to  study  the 
size  and  motion  of  these  sand-waves,  it  is  necessary  to  take 
soundings  very  near  together,  on  longitudinal  lines  over  the 
same  paths  at  frequent  intervals  for  a considerable  period. 
The  boat  is  allowed  to  drift  with  the  current  and  the  lead  floats 
with  the  boat  near  the  bottom.  It  is  lowered  to  the  bottom 
every  few  seconds  and  the  depth  and  time  recorded.  About 
once  a minute  the  boat  is  located  by  two  instruments  on  shore 
or  in  the  boat,  and  so  the  exact  path  of  the  boat  located.  A 
profile  of  the  bottom  can  then  be  drawn  for  the  path  of  the 
boat.  A few  days  later  the  same  line  is  sounded  again  in  a 
similar  manner  and  the  two  profiles  compared.  It  will  be 
found  that  the  waves  have  all  moved  down-stream  a short  dis- 
tance, the  principal  waves  still  retaining  their  main  charac- 
teristics, so  that  identification  is  certain."^ 

242.  Areas  of  Cross-section  are  obtained  by  plotting 
the  soundings  on  cross-section  paper,  the  horizontal  scale  be- 
ing about  one  tenth  or  one  twentieth  of  the  vertical.  The 
horizontal  line  representing  the  water-surface  is  drawn,  and  the 
plotted  soundings  joined  by  a free-hand  line.  The  enclosed 
area  is  then  measured  by  the  planimeter.  If  the  horizontal 
scale  is  50  feet  to  the  inch  and  the  vertical  scale  5 feet  to  the 
inch,  then  each  square  inch  of  the  figure  represents  250  square 
feet  of  area.  The  planimeter  should  be  set  to  read  the  area 
in  square  inches,  and  the  result  multiplied  by 


* It  is  believed  the  author  made  the  first  successful  study  of  the  size  and 
rate  of  motion  of  sand-waves,  at  Helena,  Ark.,  on  the  Mississippi  River,  in 
1879.  Rep.  Chief  of  Engineers,  U.  S.  A.,  1879,  vol.  iii.,  p.  1963. 
f See  p.  143  for  a description  and  theory  of  the  planimeter. 


HYDROGRAPHIC  SUR  VE  YING. 


291 


Areas  of  cross-section  are  usually  taken  in  running  water, 
and  here  great  care  must  be  taken  to  get  vertical  soundings, 
and  to  make  the  proper  sounding-line  corrections.  They 
should  be  taken  near  enough  together  to  enable  the  bottom 
line  to  be  drawn  with  sufficient  accuracy. 

BENCH-MARKS,  GAUGES,  WATER-LEVELS,  AND  RIVER-SLOPE. 

243.  Bench-marks  should  be  set  in  the  immediate  vi- 
cinity of  each  water-gauge,  and  these  connected  by  duplicate 
lines  of  levels  with  the  reference-plane  of  the  survey.  If  the 
gauge  is  not  very  firmly  set,  or  if  it  is  necessary  to  move  it  for 
a changing  stage,  its  zero  must  be  referred  again  to  its  bench- 
mark by  duplicate  levels,  whenever  there  is  reason  to  suspect 
it  may  have  been  disturbed.  Such  bench-marks  as  these  are 
usually  spikes  in  the  roots  of  trees  or  stumps. 

244.  Water-gauges  are  of  various  designs,  according  to 
the  situation  and  the  purpose  in  view.  For  temporary  use 
during  the  period  of  a survey,  a staff  gauge  is  best,  consisting 
of  a board  painted  white,  of  sufficient  length,  graduated  to  feet 
and  tenths  in  black.  Sometimes  it  is  graduated  to  half-tenths, 
but  this  is  useless  unless  in  still  water,  and  there  is  never  any 
need  of  graduation  finer  than  this.  The  gauge  maybe  read  to 
hundredths  of  a foot  if  the  water  is  calm  enough.  It  should 
be  nailed  to  a pile  or  to  a stake  driven  firmly  near  the  water’s 
edge.  It  is  read  twice  a day,  or  oftener,  if  the  needs  of  the 
service  require. 

For  the  continuous  record  of  tidal  stages  an  automatic,  or 
self-registering,  gauge  is  employed.  For  rivers  with  widely 
varying  stage  an  inclined  scantling  is  fixed  to  stakes  set  from 
low  to  high  water  along  up  the  sloping  bank.  It  should  be 
placed  at  a point  where  the  bank  is  neither  caving  away  nor 
growing  by  filling-in  of  new  deposits.  After  the  scantling  is 
set  (the  slopes  not  necessarily  the  same  throughout  its  length), 
the  foot  and  tenth  graduations  are  set  by  means  of  a level  and 


292 


S UK  VE  Y INC. 


marked  by  driving  copper  tacks.  The  automatic  gauge  is 
described  in  Chap.  XIV.  ’ The  staff  gauge  is  the  one  generally 
used  for  engineering  and  surveying  purposes. 

245.  Water-levels. — The  surface  of  still  water  is  by  defi- 
nition a level  surface.  This  fact  is  used  to  great  advantage 
on  the  sea-coast,  on  lakes,  ponds,  and  even  on  streams  of  little 
slope  or  on  such  as  have  a known  slope.  Thus,  in  finding  the 
elevations  of  the  Great  Lakes  above  the  sea-level,  the  elevation 
above  mean  tide-water  of  the  zero  of  a certain  water-gauge  at 
Oswego,  N.  Y.,  on  Lake  Ontario,  was  determined.  Then  the  rela- 
tive elevations  of  the  zeros  of  certain  gauges  at  Ports  Dalhousie 
and  Colborne,  at  the  lower  and  upper  ends  of  the  Welland 
Canal  respectively,  were  found  by  levelling  between  them,  thus 
connecting  Lake  Ontario  with  Lake  Erie.  Lakes  Erie  and  Hu- 
ron were  joined  in  a similar  manner  by  connecting  a gauge  at 
Rockwood,  at  the  mouth  of  the  Detroit  River  with  one  at  Lake- 
port,  at  the  lower  end  of  Lake  Huron.  Lakes  Michigan  and 
Huron  were  assumed  to  be  of  the  same  level  on  account  of 
the  small  flow  between  them  and  the  very  large  sectional  area 
of  the  Straits  of  Mackinac.  F'inally,  a gauge  at  Escanaba,  on 
Lake  Michigan,  was  joined  by  a line  of  levels  with  one  at  Mar- 
quette, on  Lake  Superior.  This  completed  the  line  of  levels 
from  New  York  to  Lake  Superior,  when  sufficient  gauge-read- 
ings had  been  obtained  to  enable  water- levels  \.o  be  carried  from 
Oswego  to  Port  Dalhousie,  on  Lake  Ontario  ; from  Port  Col- 
borne to  Rockwood,  on  Lake  Erie  ; and  from  Lakeport,  on 
Lake  Huron,  to  Escanaba,  on  Lake  Michigan.  It  was  found 
that  these  water-levels  were  very  accurate.  Relative  gauge- 
readings  were  compared  for  calm  days,  as  well  as  for  days 
when  the  wind  was  in  various  directions,  and  a final  mean 
value  found  which  in  no  case  had  a probable  error  as  great  as 
o.i  foot.* 


* See  Primary  Triangulatior.  of  the  U.  S.  Lake  Survey. 


H YDROGRA PHIC  S UR  VE  YING. 


293 


A line  of  levels  run  along  a lake  shore  or  canal  in  calm 
weather  should  be  checked  at  intervals  by  reading  to  the 
water-surface,  and  in  a topographical  survey  the  stadia-rod 
should  frequently  be  held  at  the  water-surface,  even  when  the 
body  of  water  is  a stream  with  considerable  slope,  as  it  gives 
a check  against  large  errors  even  then,  and  at  the  same  time 
gives  the  slope  of  the  stream.  Mean  sea-level  at  all  points 
on  the  sea-coast  is  universally  assumed  to  define  one  and  the 
same  level  surface.  It  is  probable,  however,  that  this  is  not 
strictly  true.  Wherever  a constant  ocean  current  sets  stead- 
ily against  a certain  coast,  it  would  seem  that  the  water  here 
must  be  raised  by  an  amount  equal  to  the  head  necessary  to 
generate  the  given  last  motion.  If  the  current  flows  into  an 
enclosed  space,  as  the  equatorial  current  into  the  Gulf  of 
Mexico,  or  the  tides  into  the  Bay  of  Fundy,  the  water-surface 
may  rise  much  higher.  There  is  some  evidence  that  the  ele- 
vation of  mean  tide  in  the  Gulf  of  Mexico  is  two  or  three  feet 
higher  than  that  of  the  Atlantic  at  Sandy  Hook.*  The  evi- 
dence on  this  point  is  as  yet  insuflicient  to  warrant  any  certain 
conclusion,  however. 

246.  River  Slope  is  a very  important  part  of  a river  survey. 
Sometimes  it  is  desirable  to  determine  it  for  a given  stretch 
of  river  with  great  care,  in  which  case  it  is  well  to  set  gauges 
at  the  points  between  which  the  slope  is  to  be  found  and  con- 
nect them  by  duplicate  lines  of  accurate  levelling.  The  gauges 
are  then  read  simultaneously  every  five  minutes  for  several 
hours  and  the  comparison  made  between  their  mean  readings. 
This  is  always  done  in  connection  with  the  measurement  of  the 
discharge  of  streams  when  the  object  is  to  find  what  function 
the  discharge  is  of  the  slope.  It  is  now  known,  however,  that 
in  natural  channels  the  discharge  is  no  assignable  function  of 


* See  paper  by  Prof.  Hilgard,  Supt.  U.  S.  C.  and  G.  Survey,  in  Trans.  Am. 
Asso.  Adv.  Sciences,  1884,  p.  446. 


294 


SUR  VE  YING. 


the  slope,  as  is  explained  in  section  259.  For  ordinary  purposes 
the  river  slope  may  be  determined  with  sufficient  accuracy  by 
simply  reading  the  level  or  the  stadia-rod  at  water-surface  as 
the  survey  proceeds,  daily  readings  of  stage  being  made  at 
permanent  gauges  at  intervals  of  fifty  miles  or  less  along  the 
river. 

In  all  natural  channels  the  local  slope  is  a very  variable 
quantity.  It  is  frequently  negative  for  short  distances  in  cer- 
tain stages,  and  over  the  same  short  stretch  of  river  it  may 
vary  enormously  at  different  stages,  and  even  for  the  same 
stage  at  different  times.  It  is  determined  by  the  local  channel 
conditions,  and  these  are  constantly  changing  in  streams  flow- 
ing in  friable  beds  and  subject  to  material  changes  of  stage. 
Great  caution  must  therefore  be  exercised  in  introducing  it 
into  any  hydraulic  formulae  for  natural  channels.  It  is  usually 
expressed  as  a fraction,  being  really  the  natural  sine  of  the 
angle  of  the  surface  to  the  horizon.  That  is,  if  the  slope  is  one 
foot  to  the  mile  it  is  = 0.000189. 

THE  DISCHARGE  OF  STREAMS. 

247.  Measuring  Mean  Velocities  of  Water  Currents. 

— This  is  usually  done  only  for  the  purpose  of  obtaining  the 
discharge  of  the  stream  or  channel,  but  sometimes  it  is  done 
for  other  purposes,  as  for  the  location  of  bridge  piers  or  harbor 
improvements.  In  the  case  of  bridge  piers  the  direction  of 
the  current  at  different  stages  must  be  known,  so  that  the  piers 
may  be  set  parallel  to  the  direction  of  the  current.  For  find- 
ing the  discharge  of  the  stream  or  other  channel  the  object  may 
be  : 

(1)  To  obtain  an  approximate  value  of  the  discharge  at  the 
given  time  and  place. 

(2)  To  obtain  an  exact  value  of  the  discharge  at  the  given 
time  and  place. 

(3)  To  obtain  a general  formula  from  which  to  obtain  sub- 


H YDROGRAPHIC  SUR  VE  YING. 


295 


sequent  discharges  at  the  given  place,  or  to  test  the  truth  of 
existing  formulae,  or  to  determine  the  relative  efficiency  of 
certain  appliances  or  methods. 

It  will  be  assumed  that  the  second  object  is  the  one  sought, 
and  modified  forms  of  the  methods  used  to  accomplish  this 
may  be  chosen  for  other  cases. 

The  mean  velocity  of  a stream  is  by  definition  the  total  dis- 
charge in  cubic  feet  per  second  divided  by  the  area  of  the 
cross-section  in  square  feet.  This  gives  the  mean  velocity  in 
feet  per  second.  Evidently  this  is  the  mean  of  the  veloci- 
ties of  all  the  small  filaments  (as  of  one  square  inch  in  area)  on 
the  entire  cross-section.  If  the  velocities  of  these  filaments 
could  be  simultaneously  and  separately  observed  and  their 
mean  taken,  this  would  be  the  mean  velocity  of  the  stream.  It 
is  quite  impossible  to  do  this  ; but  the  nearer  this  is  approached, 
the  more  accurate  is  the  final  result.  If,  however,  we  could 
obtain  by  a single  observation  the  mean  velocity  of  all  the  fila- 
ments in  a vertical  plane,  the  number  of  necessary  observations 
would  be  diminished  without  diminishing  the  accuracy  of  the 
result.  There  are  two  common  methods  of  measuring  the  ve- 
locities of  filaments  at  any  part  of  the  cross-section,  and  one 
for  obtaining  at  once  the  mean  velocity  in  a vertical  plane. 
These  are  by  sub-surface  floats  and  current-meters,  and  by  rod 
floats,  respectively. 

248.  By  Sub-surface  Floats. — The  ideal  sub-surface  float 
consists  of  a large  intercepting  area  maintained  at  any  depth 
in  a vertical  position  by  means  of  a fine  cord  joined  to  a sur- 
face float  of  minimum  immersion  and  resistance,  which  bears 
a signal-flag.  As  good  a form  as  any,  perhaps,  for  the  lower 
float,  or  intercepting  plane,  consists  of  two  sheets  of  galvanized 
iron  set  at  right  angles,  and  intersecting  in  their  centre  lines,  as 
shown  in  Fig.  73.  There  are  cylindrical  air-cavities  along  the 
upper  edges  and  lead  weights  attached  to  the  lower  edges  of 
the  vanes.  These  serve  to  give  the  desired  tension  on  the 


296 


S UK  VE  YING. 


connecting  cord  and  to  maintain  the  float  in  an  upright  posi- 
tion, even  though  the  cord  is  drawn  out  of  the  vertical  by 
faster  upper  currents.  Tlie  vanes  should  be  from  six  to  fifteen 
inches  in  breadth  by  from  eight  to  twenty  inches  high,  accord- 
ing to  the  size  of  the  stream.  The  circular  ribs  serve  simply 
to  hold  the  vanes  in  place.  The  upper  float  is  hollow,  cylin- 


Fig.  73. 


drical  in  plan,  and  carries  a small  flag.  The  tension  on  the 
cord  should  be  from  one  to  five  pounds,  according  to  the  size 
of  the  floats.  The  cord  itself  should  be  of  woven  silk  and  as 
small  as  possible,  so  as  to  exercise  a minimum  influence  on  the 
motion  of  the  lower  float.  Wire  is  not  suitable  for  this  pur- 
pose, as  it  kinks  badly  in  handling.  The  theory  is  that  the 
lower  float  will  move  with  the  water  which  surrounds  it,  and 
that  the  upper  float  will  be  accelerated  or  retarded  according 


H YDROGRAPHIC  SUR  VE  YING. 


297 


as  the  surface  current  is  slower  or  faster  than  that  at  the  sub- 
merged float.  The  velocity  of  the  current  at  any  depth  can 
thus  be  determined  by  running  the  lower  float  at  this  depth 
and  observing  the  time  required  for  the  upper  float  to  pass 
between  two  fixed  range-lines  at  right  angles  to  the  direction 
of  the  current  about  two  hundred  feet  apart.  The  floats  are 
started  about  one  hundred  feet  above  the  upper  range-line,  and 
picked  up  after  having  passed  the  lower  range.  Two  transits 
are  usually  used  for  locating  and  timing  the  floats,  one  being 
set  on  each  range.  When  the  float  approaches  the  upper 
range  the  observer  on  this  line  sets  his  telescope  on  range  and 
calls  “ ready”  as  the  float  enters  his  field  of  view.  The  other 
observer  then  clamps  his  instrument  and  follows  the  float  with 
the  aid  of  the  slow-motion  or  tangent  screw.  When  the  float 
crosses  the  vertical  wire  of  the  upper  instrument  he  calls  “ tick,” 
and  the  lower  observer  reads  his  horizontal  angle.  He  then 
sets  his  telescope  on  the  lower  range  while  the  upper  observer 
follows  the  float  with  his  telescope,  and  the  operation  is  re- 
peated to  obtain  an  intersection  on  the  lower  range.  One  or 
two  timekeepers  are  needed  to  note  the  time  of  the  two 
'‘tick”  calls,  the  difference  being  the  time  occupied  by  the 
float  in  passing  from  the  upper  to  the  lower  range-line.  Both 
these  signals  are  sometimes  transmitted  telegraphically  to  a 
single  timekeeper.  When  the  angles  are  plotted  the  path  of 
the  float  is  also  obtained. 

If  the  channel  is  not  too  wide,  wires  may  be  stretched 
across  the  stream  and  the  float  stations  marked  on  these,  or 
the  float  stations  may  be  determined  by  means  of  fixed  ranges 
on  shore.  The  passage  of  the  floats  across  the  section  lines 
may  then  be  noted  by  a single  individual  without  a transit, 
using  a stop-watch  and  possibly  a field-glass.  He  starts  the 
watch  when  the  float  reaches  the  upper  section,  walks  to  the  * 
lower  section,  and  stops  the  watch  when  the  float  passes  this 
range-line.  The  near  range  consists  of  a plumb-line,  or  wire 


2g8 


SUR  VE  YING. 


suspended  vertically;  and  the  observer  stands  several  feet  back 
of  this,  and  brings  it  in  line  with  the  range-post  on  the  opposite 
side  of  the  stream. 

If  several  floats  are  started  a few  minutes  apart  at  the  same 
station  and  at  the  same  depth,  they  will  sometimes  vary  as 


Fig.  74. 


’■nuch  as  twenty  per  cent  in  their  times  of  passage,  showing 
great  irregularity  in  the  velocity  of  different  parts  of  the  same 
filament.  This  is  due  to  internal  movements  in  the  water, 
such  as  “ boils,”  eddies,  etc.  It  is  for  this  reason  that  great 
refinement  in  such  observations  is  useless.  A float  observation 


H YDROGRA  PHIC  SUR  VE  YING. 


299 


Fig.  75. 


gives  only  the  velocity  of  a given  small  volume  of  water  which 
surrounds  the  lower  float,  while  a current-meter  observation, 
as  will  be  seen,  gives  the  mean  velocity  of  a given  jilamejit  of 


300 


SUJ^  VE  YING. 


the  stream  of  any  required  length.  And  as  different  portions 
of  the  same  filament  have  very  different  longitudinal  velocities, 
it  requires  a great  many  float  observations  to  give  as  valuable 
information  as  may  be  obtained  by  running  a current-meter 
in  the  same  filament  for  one  minute. 

If  discharge  observations  are  to  be  repeated  many  times  at 
the  same  sections,  then  an  auxiliary  range  should  be  established 
from  which  to  start  the  floats;  and  if  it  is  desirable  to  always 
run  them  over  the  same  paths,  these  may  be  fixed  by  means 
of  a system  of  intersecting  ranges  as  described  on  p.  289. 

249.  By  Current-meter. — This  is  the  most  accurate  method 
of  obtaining  sub-surface  velocities  ever  yet  devised.  Three 
patterns  of  current-meters  are  shown  in  Figs.  74  and  75. 
The  first  and  third  are  shown  in  elevation,  together  with  the 
electrical  recording-apparatus.  The  second  is  shown  in  plan. 
The  first  has  helicoidal  and  the  other  two  conical  cup-shaped  * 
vanes.  Neither  has  any  gearing  under  water,  the  record  being 
kept  by  means  of  an  electrical  circuit  which  is  made  and  bro- 
ken one  or  more  times  each  revolution.  The  cup  vanes  are 
better  adapted  to  water  carrying  fibrous  materials  which  tend 
to  collect  on  the  moving  parts.  The  friction  can  also  be  made 
less  on  the  cup  meters,  agate  or  iridium  bearings  being  used. 
The  recording-apparatus  is  kept  on  shore  or  in  a boat,  while 
the  meter  is  suspended  by  proper  appliances  at  any  point  of 
the  section  at  which  the  velocity  of  the  current  is  to  be  measured. 
In  deep  water  a boat,  or  catamaran,  is  anchored  at  the  desired 


* Invented  by  Gen.  Theo.  G.  Ellis,  and  first  used  on  the  survey  of  the 
Connecticut  River.  The  telegraphic  attachment  is  due  to  D.  Farrand  Henry 
of  Detroit,  Mich.  See  Report  of  the  Chief  of  Engineers,  U.  S.  A.,  1878,  p.  308. 

The  form  shown  in  Fig.  75  is  due  to  W.  G.  Price,  and  was  specially  de- 
signed to  be  used  on  the  Mississippi  River.  It  is  very  strong  and  well  pro- 
tected against  floating  drift.  The  first  two  forms  are  manufactured  by  Buff 
& Berger,  of  Boston,  while  the  Price  meter  is  made  by  W.  & L.  E.  Gurley, 
Troy. 


H YDROGRA  PHIC  S UR  VE  YING. 


301 


point,  and  a weight  attached  to  the  meter,  which  is  then  lowered 
to  the  requisite  depth  by  means  of  a windlass.  After  it  is  in 
place  the  connection  is  made  with  the  battery,  and  the  record 
kept  for  a given  period  of  time,  as  for  two  or  three  minutes. 
If  the  operation  is  to  be  repeated  often  at  the  same  section  a 
wire  anchorage  laid  across  the  stream  above  the  line  would  be 
found  useful.  This  wire  is  anchored  at  intervals  and  is  used 
both  for  holding  the  boat  (or  catamaran)  in  place  and  for  pull- 
ing it  back  and  forth  across  the  stream.  In  large  rivers  a 
steam-launch  may  be  required  for  handling  the  catamaran.* 
In  this  case  the  record  begins  and  ends  when  the  observer  is 
brought  on  range,  it  being  impossible  to  hold  up  steadily 
against  the  current.  If  only  the  discharge  of  the  stream  is 
sought,  the  meter  is  run  at  mid-depth  at  a sufficient  number 
of  points  in  the  section. 

The  mean  velocity  in  a vertical  section  at  a given  point  may 
be  obtained  by  moving  the  meter  at  a uniform  rate  from  sur- 
face to  bottom  and  back  again,  noting  the  reading  of  the  regis- 
ter for  the  two  surface  positions,  and  also  for  the  bottom  posi- 
tion. If  the  boat  was  stationary  and  the  rates  of  lowering  and 
raising  strictly  constant  and  equal,  the  number  of  revolutions  in 
descending  and  in  ascending  should  be  equal.  Either  of  these 
registrations,  divided  by  the  time,  would  give  the  mean  regis- 
tration per  second  of  all  the  filaments  in  that  vertical  plane. 
The  mean  of  the  downward  and  upward  results  may  be  used 
as  giving  the  mean  velocity  in  that  vertical  plane.  This  will 
not  be  quite  accurate,  since  it  is  impossible  to  run  the  meter 
very  close  to  the  bottom,  but  the  results  will  be  found  useful 
for  comparison  with  the  mid-depth  results.  Such  observations 
are  sometimes  called  integrations  in  a vertical  plane. 

250.  Rating  the  Meter. — When  any  kind  of  current-meter 


* For  a description  of  the  latest  methods  used  in  gauging  the  Mississippi 
River  see  Report  of  the  Miss.  Riv.  Com.  for  1883,  Appendix  F. 


302 


sc//!  VE  YING. 


is  used  for  determining  the  velocity  of  passing  fluids,  only  the 
number  of  revolutions  of  the  wheel  carrying  the  vanes  is  ob- 
served for  a given  time.  Before  the  velocity  of  the  fluid  in  feet 
per  second  can  be  found,  the  relation  between  the  rate  of  revo- 
lution of  the  wheel  and  the  rate  of  motion  of  the  fluid  must  be 
determined  for  all  velocities  that  are  to  be  observed.  The  de- 
termination of  this  relation  is  called  rating  the  meter.  It  is  usu- 
ally done  by  causing  the  meter  to  move  through  still  water  at 


a uniform  speed,  and  noting  the  time  occupied  and  the  corre- 
sponding number  of  registrations  made  in  passing  over  a given 
distance.  It  may  be  attached  to  the  prow  of  a boat,  as  shown 
in  Fig.  76,  the  electric  register  being  in  the  boat.  The  dis- 
tance divided  by  the  time  gives  the  rate  of  motion  or  velocity 
of  the  meter  through  the  water.  The  number  of  registrations 
(revolutions  of  the  wheel)  divided  by  the  time  gives  the  rate 
of  motion  of  the  wheel.  The  ratio  of  these  two  rates  is  the 
coefficient  by  which  the  registrations  of  the  meter  are  trans- 
formed into  the  velocity  of  the  current.  This  ratio  is  not  a 
constant,  but  is  usually  a linear  function  of  the  velocity.  Thus, 
if  the  observations  be  plotted,  taking  the  number  of  registra- 
tions per  second  as  abscissae  and  the  velocities  in  feet  per 
second  as  ordinates,  they  will  be  found  to  fall  nearly  in  a right 
line,  the  equation  of  which  is 


y — ax-\-b  . . . . (i) 


H YDROGRA  PHIC  SUR  VE  YING. 


303 


Here  x and  j/  are  the  observed  quantities,  while  a and  b are 
constants  for  the  given  instrument.  If  these  constants  could 
be  found,  then  the  values  of  y (velocity)  could  be  obtained  for 
all  observed  values  of  x (registrations).  There  are  two  ways  of 
solving  this  problem — one  graphical  and  one  analytical.  Evi- 
dently any  two  observations  at  different  speeds  would  give 
values  of  a and  b\  but  to  find  the  best  or  most  probable  values 
of  these  constants  a great  many  observations  are  taken,  so  that 
we  have  many  more  observations  than  we  have  unknown  quan- 
tities. Each  pair  of  observations  would  give  a different  set  of 
values  of  a and  b.  The  most  convenient  method  of  finding 
the  most  probable  values  of  these  functions,  though  somewhat 
approximate,  is 

(i)  The  Graphical  Method  of  Solution. — This  consists  simply 
in  plotting  the  corresponding  values  of  x and  y on  coordi- 
nate paper,  and  drawing  the  most  probable  straight  line  through 
the  points.  Then  the  tangent  of  the  angle  this  line  makes 
with  the  axis  of  x is  and  the  intercept  on  the  axis  of  y is  b. 
One  point  on  this  most  probable  line  is  the  point  {x^y^,  and 
y^  being  the  mean  values  of  the  coordinates  of  all  the  plotted 
points.  This  is  shown  by  equation  (3).  Having  determined 
this  point,  a thread  may  be  stretched  through  it  and  swung 
until  it  seems  to  be  in  a position  of  equilibrium,  when  each 
point  is  conceived  as  an  attractive  force  acting  on  the  line,  the 
measure  of  the  force  being  the  vertical  intercept  between  the 
point  and  the  line.  The  arms  of  these  forces  are  evidently 
their  several  abscissae.  Or  the  forces  may  be  measured  by 
their  horizontal  intercepts,  and  then  their  arms  are  their  seve- 
ral ordinates.  For  the  position  of  equilibrium  the  sum  of  the 
moments  of  these  forces  about  the  point  {x^y^)  would  be  zero.* 
Such  a determination  of  a and  b would  be  found  sufficiently 
accurate  for  all  practical  purposes,  but  if  desired  the  problem 
may  be  solved  by 

* All  this  simply  means  to  fix  this  most  probable  line  by  eye,  through  the 
point  (jfo  ya),  giving  greatest  weight  to  the  extreme  points. 


304 


SURVEYING. 


(2)  The  Rigid  or  Analytical  Method. — Equation  (i)  may  be 
written 


b xa  — y — o. 

Every  observation  may  be  written  in  this  form,  these  being 
called  the  observation  equations.  It  is  probable  that  no  given 
values  of  a and  b would  satisfy  more  than  two  of  these  obser- 
vations ; and  if  the  most  probable  values  be  used,  there  would, 
in  general,  be  no  single  equation  exactly  satisfied.  If  we  let 
;r„  etc.,  etc.,  and  2/,,  2/,,  etc.,  be  the  several  values  of 

.r,  j/,  and  the  corresponding  residuals  for  the  several  observa- 
tion equations,  we  would  have 

h-\- x,a— 

b x,a  - y,- v,\  (2) 

b-^x^a  —y„  = 

Since  b enters  alike  in  all  of  them,  it  is  evident  that  these 
equations  are  all  of  equal  value  for  determining  b.  Also,  since 
the  properly  weighted  arithmetic  mean  is  the  most  probable 
value  of  a numerously  observed  quantity,  and  since  in  this  case 
the  equations  (or  observations)  have  equal  weight  for  deter- 
mining b,  we  may  form  from  the  given  series  of  equations  a 
single  standard  or  “ normal " equation  which  will  be  the  arith- 
metic mean  of  the  observation  equations  ; put  this  equal  to  zero 
and  say  this  shall  give  the  value  of  b.  If  x^  and  y^  be  the  mean 
values  of  the  observed  x*s  and^s,  we  would  then  have,  by  add- 
ing the  equations  all  together  and  dividing  by  their  number 


b + x,a—y,  = o,orb=y,—x,a.  ...  (3) 


HYDROGRAPHIC  SURVEYING. 


305 


Substituting  this  value  of  b in  equation  (2),  we  have 


{x,  — x,)a  — (f,  -jy,)  = v,; 
(x,  - x,)a  - {y,  - 7.)  = V, : 


• • (4) 


{x„  — x,)a  — (jj/„  —y„)  = v„. 


We  here  have  a series  of  equations  involving  one  unknown 
quantity ; but  they  evidently  are  not  of  equal  value  in  deter, 
mining  the  unknown  quantity  a,  since  its  coefficients  are  very 
different.  In  fact,  the  relative  value  of  these  equations  for  de- 
termining a is  in  direct  proportion  to  the  size  of  this  coefficient, 
so  that  if  this  coefficient  is  twice  as  large  in  one  equation  as  in 
another,  the  former  equation  has  twice  the  value  of  the  latter 
for  determining  a.  In  other  words,  they  should  all  be  weighted 
in  proportion  to  the  values  of  these  coefficients,  and  a conve- 
nient way  of  doing  this  is  to  multiply  each  equation  through 
entire  by  this  coefficient.  The  resulting  multiplied  equations 
then  have  equal  weight,  and  may  then  be  added  together  to 
produce  another  ‘‘  normal  equation  for  finding  a.  This  result- 
ing equation  is 


lix-x.yy-lix-x,  {y-y,)\=o,  ...  (5) 

where  [ ] is  a sign  of  summation.  If  we  had  divided  this 
equation  by  the  number  of  observation  equations  m,  it  would 
in  no  sense  have  changed  it  so  far  as  the  value  of  a is  concerned. 
From  equation  (5)  we  can  find  the  mean  or  most  probable 
value  of  a,  which  when  substituted  in  (3)  gives  the  most  prob- 
able value  of  b.  These  values  should  agree  very  closely  with 
those  found  by  the  graphical  method.  The  analytical  method 
here  given  is  precisely  that  by  least  squares,  though  arrived  at 
through  the  conception  of  a properly  weighted  arithmetic  mean, 
instead  of  by  making  the  sum  of  the  squares  of  the  residuals  a 
minimum. 


20 


3o6 


SURVEYING. 


The  following  is  an  actual  example  from  the  records  of  the 
Mississippi  l-^ver  Survey: 


REX)UCTION  OF  OBSERVATIONS  FOR  RATING  METER  A, 
taken  at  Paducah,  Ky.,  June  21,  1882. 

W.  G.  Price,  Observer.  L.  L.  Wheeler,  Computer.* 


No. 

r 

t 

jr 

y 

X — JTo 

y-yo 

(x  - JTo)’ 

(.r  — j-q) 

^y -y<i) 

Remarks.^ 

I 

100 

53 

I 886 

3-774 

+ O.II7 

+ 0.245 

-f-  0.014 

-j-  0.029 

Observations 

2 

1 

lOI 

44 

2.295 

4-544 

+ 0.526 

+ I. 015 

+ 0.277 

+ 0.534 

made  with 

3 

^OI 

41 

2.464 

4.878 

+ 0.695 

-h  1.349 

+ 0.483 

4-  0.938 

meter  on  vertical 

4 

96 

124 

0 774 

1.613 

- 0.995 

— 1.916 

+ 0.990 

4"  I . 906 

iron  rod,  five 

5 

94 

152 

0.618 

1.316 

- I.I5I 

— 2.213 

+ 1-325 

4-  2.548 

feet  in  front  of 

6 

90 

193 

0.466 

1.036 

- I.. 303 

- 2.493 

+ 1-^97 

+ 3.249 

bow  of  skiff,  in 

7 

91 

181 

0.503 

1. 105 

- 1.266 

- 2.424 

+ 1.603 

4-  3-069 

pond. 

8 

103 

28 

3.678 

7.142 

+ 1-909 

+ 3.613 

-h  3-644 

4-  6.903 

9 

100 

53 

1.886 

3-774 

O.II7 

+ 0.245 

-f-  0.014 

4-  0.029 

Length  of  base 

10 

98 

73 

1.342 

2.740 

- 0.427 

— 0.789 

-}-  0.182 

4-  0.337 

= 200  feet. 

II 

103 

29 

3-552 

6.896 

+ 1.783 

+ 3-367 

4-  3-178 

4-  6.002 

w = 

O-o  = 

19.464 

1.769 

38.818 

3-529 

= /0 

= 13-407 

25.544= 

-[{x-x^){y-y^)'\ 

Normal  Equations. 

b -f-  1.769a  — 3529  = o ; Whence  a — 1.905  ; 

13.407a  — 25.544  = o.  ^=0.159. 

Equation  for  Rating, 
y = 1.905;!;  -f  0.159. 

Even  where  the  analytical  method  is  to  be  used  it  is  al- 
ways well  to  plot  the  observations  for  purposes  of  study. 
Then  if  any  observations  are  especially  discrepant,  the  fact  will 
appear.  By  consulting  column  six  of  the  computation  it  will 

* In  the  original  computation  the  method  by  least  squares  was  used  and  the 
probable  errors  of  a and  b found. 


H YDROGRAPHIC  S UR  VE  YING. 


307 


be  seen  that  observations  of  greatest  weight  were  those  taken  at 
very  high  and  at  very  low  velocities.  If  the  observations  were 
taken  in  three  groups  about  equally  spaced,  an  equal  number 
of  observations  in  each  group,  the  members  of  a group  being 
near  together,  then  the  mean  of  each  group  could  be  used  as 
a single  observation.  The  middle  group  would  serve  to  show 
whether  or  not  the  unknown  quantities  were  linear  functions  of 
each  other,  since,  if  they  were,  the  three  mean  observations 
should  plot  in  a straight  line.  The  value  of  a could  be  com- 
puted from  the  two  extreme  mean  observations,  and  the  value 
of  b from  the  mean  of  all  the  observations  as  before.  This 
would  give  a result  quite  as  accurate  as  to  treat  them  separately. 

If  the  observations  do  not  plot  in  a straight  line,  draw 
the  most  probable  line  through  them,  and  prepare  a table  of 
corresponding  values  of  x and  j/  from  this  curve.  In  any  case, 
a reduction  table  should  be  used. 

The  meter  should  be  rated  frequently  if  accurate  results  are 
required.  In  the  rating  the  meter  should  be  fastened  several 
feet  in  front  of  the  bow  of  the  boat,  and  in  its  use  it  should  be 
run  at  a sufficient  distance  from  the  boat  or  catamaran  to  be 
free  from  any  disturbing  influence  on  the  current. 

251.  By  Rod  Floats. — These  may  be  either  wooden  or  tin 
rods,  of  uniform  size,  loaded  at  the  bottom,  and  arranged  for 
splicing  if  they  are  to  be  used  in  deep  water.  If  the  channel 
were  of  uniform  depth,  and  the  rod  reached  to  the  bottom  with- 
out actually  touching,  then  the  velocity  of  the  rod  would  be 
the  mean  velocity  of  all  the  filaments  in  that  vertical  plane,* 


* This  is  not  strictly  true,  since  the  pressure  of  a fluid  upon  a body  moving 
through  it  varies  as  the  square  of  its  relative  velocity.  The  rod  moves  faster 
than  the  bottom  filaments  and  slower  than  the  upper  filaments,  but  this  differ- 
ence Is  greatest  at  the  bottom.  Therefore,  the  retarding  action  of  the  bottom 
filaments  will  have  undue  weight,  as  it  were,  and  so  the  velocity  of  the  rod  will 
really  be  about  one  per  cent  slower  than  the  mean  velocity  of  the  current.  See 
“Lowell  Hydraulic  Experiments,”  by  James  B.  Francis. 


3o8 


SUJ^  VE  Y I NG. 


and  this  is  the  value  sought.  In  practice  the  rod  can  never 
reach  the  bottom,  even  in  smootli,  artificial  channels,  while  in 
natural  channels  the  irregularities  are  usually  such  as  prohibit 
its  use  within  several  feet  of  the  bottom.  The  methods  of 
observation  are  the  same  as  with  the  double  floats,  and  their 
velocity  is  the  mean  velocity  of  the  water  in  that  plane  to  the 
depth  of  immersion.  I'or  artificial  channels,  and  for  natural 
channels  not  more  than  twenty  or  thirty  feet  deep,  rod  floats 
may  be  advantageously  used.  Beyond  that  depth  they  cannot 
be  made  of  sufficient  length  to  give  reliable  results.  The 
method  is,  therefore,  best  adapted  to  artificial  channels  of  uni- 
form cross-section. 

The  immersion  of  the  rod  should  be  at  least  nine  tenths  of 
the  depth  of  the  water,  in  which  case,  and  for  uniform  channels, 
as  wooden  flumes,  Francis  found  that  the  velocity  of  the  rod 
required  the  following  correction  to  give  the  mean  velocity  of 
the  water  in  that  vertical  plane : 

Vm  = Vr  [I— 0.Il6(V7j-0.l)]. 

Where  Vm  = mean  velocity  in  vertical  plane ; 

Vr  = observed  velocity  of  rod  ; 

_ depth  of  water  below  bottom  of  rod 
depth  of  water 

For  natural  channels,  or  for  a less  immersion  than  nine- 
tenths  of  the  depth  the  formula  cannot  be  used  with  certainty. 
The  rods  should  be  put  into  the  water  at  least  twenty  feet 
above  the  upper  section. 

252.  Comparison  of  Methods. — (i)  The  method  by  double 
floats  is  adapted  to  large  and  deep  rivers,  or  rapid  currents 
carrying  much  drift  or  impeded  by  traffic.  It  may  be  used 
in  all  cases,  but  it  has  the  disadvantage  of  registering  only  the 
velocity  of  a small  volume  of  water  surrounding  the  lower 
float. 


H YDROGRAPHIC  SUR  VE  YING. 


309 


(2)  The  method  by  meters  is  adapted  to  large  or  small 
streams.  It  records  the  mean  velocity  of  a filament  of  indefinite 
length ; but  it  cannot  be  used  where  the  water  carries  consider- 
able floating  debris,  or  where  the  current  is  too  swift  to  admit 
of  a safe  anchorage. 

(3)  The  method  by  rods  is  best  adapted  to  small  channels 
of  uniform  section  ; it  records  the  mean  velocity  in  a vertical 
plane  to  a depth  equal  to  its  immersion,  and  it  can  be  univer- 
sally used  when  the  law  of  the  velocities  in  a vertical  plane  is 
known,  for  then  a proper  coefficient  could  be  derived  for  any 
depth  of  immersion. 

(4)  One  rod  observation  of  sufficient  immersion  is  prob- 
ably as  good  as  several  float  observations,  and  a current-meter 
observation  of  two  or  three  minutes  is  worth  as  much  as 
twenty  float  observations  for  the  same  filament,  provided  the 
meter’s  rate  is  constant  and  well  determined. 

(5)  The  rods  and  floats  are  cheaper  in  first  cost  than  the 
meter  ; but  if  the  work  is  to  be  prosecuted  for  a considerable 
period,  the  excess  in  the  cost  of  the  outfit  will  be  more  than 
balanced  by  the  diminished  cost  of  the  work,  by  using  the 
meter.  On  the  whole,  it  may  be  said  that  the  method  by  cur- 
rent-meter is  the  most  accurate  and  satisfactory  of  any  yet  de- 
vised for  measuring  the  velocity  of  running  water. 

253.  The  Relative  Rates  of  Flow  in  Different  Parts  of 
the  Cross-section. — (i)  In  a horizontal  plane.  If  the  cross- 
section  of  a stream  were  approximately  the  segment  of  a circle, 
then  the  relative  rates  of  flow  of  the  different  filaments  in  any 
horizontal  plane  would  be  very  nearly  represented  by  the  ordi- 
nates to  a parabola,  the  axis  of  the  parabola  coinciding  with 
the  middle  of  the  stream.  If  there  .should  be  any  shoaling  in 
any  part  of  this  ideal  section  the  corresponding  ordinates  would' 
be  shortened,  so  that  when  the  curve  of  the  bottom  is  given 
the  curve  of  velocities  in  a horizontal  plane  can  be  fairly  pre- 
dicted. This  applies  only  to  straight  reaches.  If  a portion  of 
the  section  has  a flat  bottom  line,  the  velocities  over  this  por- 


310 


SURVEYING. 


tion  will  be  about  uniform.  Where  the  depth  is  chanjrin^  rap- 
idly  on  the  section,  there  the  velocities  will  be  found  to  change 
rapidly  for  given  changes  in  positions  across  the  section. 

It  follows  from  this  that  observation  stations  should  be 
placed  near  together  where  the  section  has  a sloping  bottom 
line,  and  they  may  be  placed  farther  apart  where  the  bottom 
line  of  the  section  is  nearly  flat.  They  are  usually  put  closer 
together  near  the  bank  than  near  the  middle  ot  the  stream. 

(2)  In  a vertical  plane.  A great  deal  of  time  and  talent  has 
been  spent  in  trying  to  find  the  law  of  the  relative  rates  of  flow 


in  a vertical  plane,  but  there  is  probably  no  law  of  universal 
application.  The  curve  representing  such  rates  of  flow  will 
always  resemble  a parabola  more  or  less,  the  axis  of  which  is 
always  beneath  the  surface  except  when  the  wind  is  down 
stream  at  a rate  equal  to,  or  greater  than,  the  rate  of  the  cur- 
rent. That  is  to  say,  the  maximum  velocity  is  always  below 
the  surface  except  where  the  surface  filaments  are  accelerated 
by  a down-stream  wind,  and  it  is  generally  found  at  about  one 
third  the  depth.  The  cause  of  this  depression  of  the  filament 
of  maximum  velocity  is  partly  due  to  the  friction  of  the  air, 


HYDROGRAPHIC  SURVEYING. 


31I 


but  mostly  to  an  inward  surface  flow  from  the  sides  toward 
the  centre,  which  brings  particles  having  a slower  motion 
towards  the  middle  of  the  surface  of  the  stream.  This  inward 
surface  flow  is  probably  due  to  an  upward  flow  at  the  sides 
caused  by  the  irregularities  of  the  bank,  which  force  the  parti- 
cles of  water  impinging  upon  them  in  the  direction  of  the  least 
resistance  which  is  vertical.*  The  curves  in  Fig.  77  represent 
the  mean  vertical  curves  of  velocity  observed  at  Columbus, 
Ky.,  on  the  Mississippi  River  and  given  in  Humphreys  and 

Scak  ^ Ret 

n ( 2 _.i ^ 

etacM  ■ ■ I ‘ 


Abbot’s  Report.  The  left-hand  vertical  line  is  the  axis  of  ref- 
erence, and  the  curves  are  found  to  fall  between  the  seven-  and 
eight-foot  lines.  That  is,  the  velocity  at  all  depths  in  this 
plane  was  between  seven  and  eight  feet  per  second.  In  this 
case  double  floats  were  used,  and  it  is  probable  that  the  bottom 
velocities  were  not  very  accurately  obtained.  The  effect  of  the 
wind  is  here  shown  in  shifting  the  axis  of  the  curve.  It  is  to 


* See  paper  by  F.  P.  Stearns  before  the  Am.  Soc.  Civ.  Engrs.,  vol.  xii.  p.  331. 


312 


SUI^VEY/NG. 


be  observed  that  these  curves  all  intersect  at  about  mid-depth. 
That  is  to  say,  the  velocity  of  the  mid-depth  filament  is  not 
affected  by  wind.  This  is  why  the  mid-depth  velocity  should 
be  chosen  when  the  velocity  of  but  a single  filament  is  to  be 
measured,  and  from  this  the  mean  velocity  in  the  vertical  sec- 
tion derived.  It  has  also  been  found  that  the  mid-depth  veloc- 
ity is  very  near  the  mean  velocity,  being  from  one  to  six  per 
cent  greater,  according  to  depth  and  smoothness  of  channel. 
In  general,  for  channels  whose  widths  are  large  as  compared 
to  their  depths,  a coefficient  of  from  .96  to  .98  will  reduce 
mid-depth  velocity  to  the  mean  velocity  in  that  vertical  plane. 

In  Fig.  78*  are  shown  the  relative  velocities  in  different  parts 
of  the  Sudbury  River  Conduit  of  Boston.  The  velocity  at 
each  dot  was  actually  measured  by  the  current-meter.  The 
lines  drawn  are  lines  of  equal  velocity,  being  analogous  to  con- 
tour lines  on  a surface,  the  vertical  ordinates  to  which  would 
represent  velocities.  The  method  of  obtaining  these  velocities 
is  shown  in  Fig.  79.  is  a pivoted  sleeve  through  which  the 
meter-rod  slides  freely.  At  A there  is  a roller  fixed  to  the  rod 
which  runs  on  the  curved  tracks  a a a.  The  graduations  on 
these  tracks  fix  the  different  positions  of  the  meter,  these  be- 
ing so  spaced  that  they  control  equal  areas  of  the  cross-section. 
Integrations  were  here  taken  in  horizontal  planes  by  moving  the 
meter  at  a uniform  rate  horizontally. 

254.  To  find  the  Mean  Velocity  on  the  Cross-section. 
— It  is  evident  that  this  mean  velocity  cannot  be  directly  ob- 
served. In  fact,  it  can  only  be  found  by  first  finding  the  dis- 
charge per  second  and  then  dividing  this  by  the  total  area  of 
the  section.  That  is  to  say,  the  mean  velocity  is,  by  definition. 


* This  and  ihe  following  figure  are  taken  from  the  paper  by  F.  P.  Stearns, 
mentioned  in  foot-note  on  the  previous  page. 


H YDROGRAPHIC  SUR  VE  YING. 


313 


MBTSi  JLPPABJiTint 
nt  rmarlof  M»nhot« , 


Fig.  79. 


SURVEYING. 


3H 


The  area  of  the  section  is  found  by  means  of  properly  located 
soundings.  The  actual  velocities  of  certain  filaments  crossing 
this  section  are  then  observed,  and  the  section  subdivided  in 
such  a way  that  the  observed  velocities  will  fairly  represent 
the  mean  velocities  of  all  the  similar  filaments  (usually  mid- 
depth) in  that  subsection.  Each  observed  velocity  is  then 
reduced  to  the  mean  velocity  in  that  vertical  plane,  and  this  is 
assumed  to  be  the  mean  velocity  in  that  subsection.  These 
mean  velocities,  multiplied  by  the  areas  of  their  corresponding 
sections,  give  the  discharges  across  these  sections,  and  the  sum 
of  these  partial  discharges  is  the  total  discharge,  Q,  in  the 
above  equation.  This  may  be  shown  algebraically  as  follows: 

Let  Fa,  Fg,  etc.,  be  the  observed  velocities ; 

C the  coefficient  to  reduce  these  to  the  mean  velocity 
in  a vertical  plane ; 

A^,  etc.,  the  partial  areas  of  the  cross-section 
corresponding  to  the  observed  velocities  F„  F,, 
Fg,  etc. ; 

A the  total  area  of  the  cross-section  = A^  A,  A^^ 
etc. ; 

Gi,  2,,  Qzr  etc.,  the  partial  discharges; 

Q the  total  discharge  ; 

zf  the  mean  velocity  for  the  entire  section. 

Then  0,  = CF,A, ; 0,  = C^A,,  etc. ; 

0 = 01  + 03  + etc.  = C{A,V,  + .^,Fg  + etc.); 

v = ^ = ^{A,V,  + A,V,  + ctc.). 


and 


HYDROGRAPHIC  SUR  VE  YING. 


315 


Fig.  8q, 


3i6 


SURVEYING. 


It  has  been  here  assumed  that  observations  are  made  at  but 
one  point  in  any  vertical  plane.  The  method  is  the  same,  how- 
ever, in  any  case,  it  only  being  necessary  to  apply  such  a co- 
efficient to  the  observed  velocity  as  will  reduce  it  to  the  mean 
velocity  in  its  own  sub-area.  If  these  partial  areas  are  made 
small,  as  in  the  case  of  the  Boston  Conduit,  the  observed  ve- 
locities may  be  taken  as  the  mean  velocities  in  those  areas ; and 
if  these  areas  are  all  equal,  which  was  also  the  case  in  this  con- 
duit, then  the  mean  velocity  is  the  arithmetic  mean  of  all  the 
observed  velocities.  The  partial  and  total  areas  are  best  found 
by  means  of  the  planimeter,  the  cross-section  having  been 
carefully  plotted  on  coordinate  paper. 

255-  Sub-currents. — It  is  often  desirable  to  know  the 
direction  as  well  as  the  velocity  of  flow  beneath  the  surface. 
The  direction-meter,*  Fig.  8o,  is  designed  to  give  the  direction 
of  sub-currents,  both  horizontally  and  vertically.  A magnetic 
needle  swings  freely  until  lifted  and  held  by  a magnet  which 
is  operated  from  above.  The  vertical  direction  is  recorded  in 
a similar  manner  on  a small  index-circle  on  the  inside  of  the 
hollow  sphere  which  always  maintains  an  upright  position. 

256.  The  Flow  of  Water  over  Weirs.f — The  most  ac- 
curate mode  of  measuring  the  flow  through  small  open  channels 
is  by  means  of  weirs.  There  are  three  kinds  of  weirs  with 
which  the  engineer  may  have  to  deal  in  measuring  the  flow  of 
water, — namely,  sharp-crested  weirs,  wide-crested  weirs,  and 
submerged  weirs. 

A sharp-crested  weir  is  one  which  is  entirely  cleared  by  the 
water  in  passing  over  it,  as  in  Fig.  81.  A wide  crest  is  shown 


* Invented  by  W.  G.  Price  and  manufactured  by  Gurley,  Troy,  N,  Y. 
f The  results  given  in  this  and  the  following  article  have  been  mostly  taken 
from  a paper  by  Fteley  and  Stearns  before  the  Am.  Soc.  Civ.  Engrs.,  vol.  xii. 
(1883),  describing  experiments  made  in  connection  with  the  new  Sudbury  River 
Conduit,  Boston,  Mass.  The  paper  was  awarded  the  Norman  medal  of  that 
society. 


HYDROGRAPHIC  SURVEYING. 


317 


Fig.  84.  Fig.  83. 


SURVEYING. 


318 


in  Fig.  82,  and  its  effect  in  increasing  the  depth  on  the  weir 
for  a given  discharge.  If  the  crest  has  a width  equal  to  the 
line  ab  in  Fig,  81,  then  the  depth  on  the  weir  is  unaffected, 
while  if  it  has  a less  width,  as  in  Fig.  83,  and  if  the  air  has  not 
free  access  to  the  interveiiing  space  beneath^  the  water  will  soon 
fill  this  space,  and  the  tendency  to  vacuum  here  will  depress 
the  overflowing  sheet  of  water,  thus  diminishing  the  depth  on 
the  weir  for  a given  flow.  The  dotted  lines  in  Fig.  84  are 


those  of  normal  flow,  the  full  lines  being  the  new  positions 
assumed  as  a result  of  the  partial  vacuum  below. 

A submerged  weir  is  one  at  which  the  level  of  the  water 
below  the  weir  is  above  its  crest,  there  being,  however,  a certain 
definite  fall  in  passing  the  weir,  as  shown  in  Fig.  85.  Here 
h — d — d'  x’s,  the  fall  in  passing  the  weir. 

Velocity  of  Approach. — This  is  the  velocity  of  the  surface- 
water  towards  the  weir  at  a distance  above  the  weir  equal 
to  about  two  and  one  half  times  the  height  of  the  weir  above 
the  bottom  of  the  channel. 

End  Contractions. — These  are  the  narrowing  effects  of  the 
lateral  flow  at  the  ends  of  the  weir.  If  this  lateral  component 
of  the  flow  is  shut  off  by  a plank  extending  several  feet  up 
stream  and  from  the  water’s  surface  to  several  inches  below 
the  top  of  the  weir,  then  there  is  no  end  contraction.  This 
arrangement  gives  more  accurate  results,  as  the  correction  for 
end  contraction  involves  some  uncertainties. 


H YDROGRAPHIC  SUR  VE  YING. 


319 


Depth  of  Water  on  the  Weir. — This  is  the  principal  function 
of  the  discharge  ; it  is  the  difference  of  eleva- 
tion between  the  top  of  the  weir  and  the  surface 
of  the  water  at  a distance  above  the  weir  equal  to 
about  2\  times  the  height  of  the  weir  above  the 
bottom  of  the  channel.  Evidently  this  is  a quan- 
tity which  cannot  be  directly  measured.  The 
best  way  of  measuring  this  quantity  is  as  follows: 

At  a convenient  point  arrange  a closed  vertical 
box  which  connects  by  a free  opening  with  the 
channel  at  about  mid-depth  at  a point  some  six 
feet  above  the  weir.  The  water  will  then  stand 
in  this  box  at  its  normal  elevation,  unaffected  by 
the  slope  towards  the  weir.  The  elevation  of 
this  water-surface  is  determined  by  means  of  a 
hook-gauge^  Fig.  86,  which  consists  of  a metallic 
point  turned  upwards  and  made  adjustable  in 
height  by  means  of  a thumb-screw.  When  the 
point  of  the  hook  comes  to  the  surface  of  the 
water  it  causes  a distorted  reflection.  The  eleva- 
tion of  the  water-surface  can  be  found  in  this 
way  with  extreme  accuracy.  The  difference  of 
elevation  between  the  point  of  the  hook  and  the 
crest  of  the  weir  can  then  be  determined  with  a 
level  and  rod.  This  difference  is  //"in  the  following  formulse. 

257.  Formulae  and  Corrections. — For  a simple  sharp- 
crested  weir,  without  end  contractions  and  with  no  velocity  of 
approach,  the  discharge  in  cubic  feet  per  second  is 

Q = Z.ZiLHi-{- 0.007 L, (i) 

where  L is  the  length  of  the  weir  and  H the  depth  of  water 
upon  it,  both  measured  in  feet.  The  weir  must  have  a level 
crest  and  vertical  ends ; it  should  be  in  a dam  vertical  on  its 


Fig.  86. 


320 


SURVEYING. 


up-stream  side  ; the  water  on  the  down-stream  side  may  stand 
even  with  the  crest  of  the  weir  if  it  has  considerable  depth. 
The  error  is  not  more  than  one  per  cent  when  the  water  on  the 
down-stream  side  covers  fifteen  per  cent  of  the  weir  area,  pro- 
vided H is  then  taken  as  the  difference  in  elevation  of  the 
water-surface  above  and  below  the  weir.  In  this  case  two- 
hook  gauges  would  be  needed.  The  crest  of  the  weir  should 
be  at  a height  above  the  bottom  of  the  channel  on  the  up- 
stream side  equal  to  at  least  twice  the  depth  on  the  weir,  to 
allow  for  complete  vertical  contraction. 

The  following  corrections  apply  to  their  respective  condi- 
tions : 

For  the  velocity  of  approach,  the  depth  on  the  weir,  H in 
equation  (i),  is  to  be  increased  by  1.5//,  where  there  is  no  end 

contraction,  h being  the  head  due  to  the  velocity,  or  h = — . 

At  sea-level  this  correction  becomes 

c = ~ = 0.0234t/’ (2) 


This  is  to  be  added  to  H in  equation  (i),  v being  measured  in 
feet  per  second. 

Where  there  is  end  contraction,  the  correction  is 


_ 2.05^/* 


= 0.03 


• • (3) 


For  end  contraction,  the  length  of  the  weir,  L in  equation 
(l),  is  to  be  shortened  by  0.1//"  for  each  such  contraction.  This 
is  a mean  value,  although  it  varies  from  o.oyH  to  0.12H  for 
different  depths  on  the  weir  varying  from  i to  0.3  foot,  the 
smaller  correction  applying  to  the  greater  depth  on  the  weir. 


H YDROGRAPHIC  SUR  VE  YING. 


321 


For  wide  crests  the  correction  to  the  depth  on  the  weir  is 
sometimes  positive  and  sometimes  negative,  as  shown  in  fig- 
ures 82  and  84.  The  following  correction  is  derived  from  care- 
ful experiments : 

C = 0.2016  -[-o.2I46z£;“  — 0.1876W,  ...  (4) 

where 

C is  the  correction  to  be  added  algebraically  to  the  depth 
on  the  wide  crest  to  obtain  the  depth  on  a sharp  crest 
which  will  pass  an  equal  volume  of  water  ; 

w is  the  width  of  the  crest  ; 

y is  the  difference  between  o.Zo'jw  and  the  depth  on  the 
crest. 

If  the  crest  is  narrower  than  the  line  ab.  Fig.  81,  then  this 
correction  is  not  to  be  applied  unless  the  water  adheres  to  the 
weir  as  in.  Fig.  84. 

Up-stream  edge  of  the  weir  rounded.  If  the  up-stream  edge 
of  the  weir  is  a small  quarter-circle,  add  seven  tenths  of  its  ra- 
dius to  the  depth  on  the  weir  before  applying  the  general  weir 
formula. 

Submerged  weir.  When  the  water  on  the  down-stream 
side  rises  above  the  level  of  the  crest,  use  the  formula  for  a 
submerged  weir,  which  is 


Q — c/ (^d-j- - j Vh, (5) 

where 

Q is  the  discharge  in  cubic  feet  per  second  ; 
c is  to  be  taken  from  the  following  table,  its  value  varying 
d' 

with  ^ ; 

I is  the  length  of  the  weir  in  feet ; 

21 


322 


SUA^  V a:  yii\g. 


d is  the  depth  on  the  weir  in  feet,  measured  from  still 
water  on  the  up-stream  side  ; 

d'  is  the  depth  to  which  the  weir  is  submerged,  measured 
from  still  water  on  the  down-stream  side ; 

h is  the  fall  and  equals  d — d' . 

The  value  of  d may  be  corrected  for  velocity  of  approach 
by  formulas  (2)  and  (3).  There  is  no  known  correction  for  the 
velocity  of  discharge  below  the  weir,  and  hence  the  formula 
can  only  be  used  for  a channel  of  large  capacity  below,  as  com- 
pared with  the  discharge,  so  that  the  velocity  here  will  be  small. 

The  following  are  the  experimental  values  of  c\ 


d' 

d‘ 

c. 

d' 

d‘ 

c. 

d' 

d' 

c. 

d' 

d ■ 

c. 

O.OI 

3-330 

0.25 

3-249 

0.55 

3.100 

0.85 

3.150 

.05 

3-360 

-30 

3-214 

.60 

3.092 

.90 

3.190 

.oS 

3-372 

-35 

3. 182 

-65 

3-089 

-95 

3-247 

.10 

3-365 

.40 

3-155 

-70 

3.092 

1. 00 

3-360 

•15 

3-327 

-45 

3-131 

-75 

3.102 

.20 

3.286 

-50 

3-I13 

.80 

1 

3-122 

This  table  is  inapplicable  to  values  of  ^ less  than  0.08,  un- 
less the  air  has  free  access  to  the  space  underneath  the  sheet. 

The  method  of  measuring  discharge  by  means  of  sub- 
merged weirs  is  adapted  to  channels  having  very  small  slope. 
A fall  as  low  as  one  half  inch  will  give  reliable  results  if  it  is 
accurately  measured. 

258.  The  Miner’s  Inch. — This  is  an  arbitrary  standard 
both  as  to  method  and  as  to  volume  of  water  discharged.  It 
rests  on  the  false  assumption  that  the  volume  discharged  is 
proportional  to  the  area  of  the  orifice  under  a constant  head 
above  the  top  of  the  orifice.  Its  use  grew  out  of  the  necessities 


H YDROGRAPHIC  SUR  VE  YING. 


323 


of  frontier  life  in  the  mining  regions  of  the  West,  and  should 
now  be  discarded  in  favor  of  absolute  units.  The  miner’s  inch  is 
the  quantity  of  water  that  will  flow  through  an  orifice  one  inch 
square,  under  a head  of  from  four  to  twelve  inches,  according  to 
geographical  locality.  Even  if  the  head  above  the  top  of  the 
orifice  be  fixed,  and  a flow  of  144  miner’s  inches  be  required, 
the  volume  obtained  would  be  3.3,  4.2,  or  4.7  cubic  feet  per 
second,  according  as  there  were  144  holes  each  one  inch  square, 
one  opening  one  inch  deep  and  144  inches  long,  or  one  opening 
twelve  inches  square,  the  tops  of  all  the  openings  being  five 
inches  below  the  surface  of  the  water.  This  simply  illustrates 
the  unreliable  nature  of  such  a unit.  In  some  localities  the 
following  standard  has  been  adopted : An  aperture  twelve 
inches  high  by  twelve  and  three-quarter  inches  wide  through 
one  one-  and  one-half-inch  plank,  with  top  of  opening  six  inches 
below  the  water-surface,  is  said  to  discharge  two  hundred 
miner’s  inches.  By  this  standard  the  miner’s  inch  is  1.5  cubic 
feet  per  minute,  or  2160  cubic  feet  in  twenty-four  hours. 
Other  standards  vary  from  1.39  to  1.78  cubic  feet  per  minute.* 
When  the  miner’s  inch  can  only  be  defined  as  a certain  num- 
ber of  cubic  feet  per  minute,  it  is  evidently  no  longer  of  ser- 
vice and  should  be  abandoned.  The  method  by  weirs  is  more 
accurate,  and  could  almost  always  be  substituted  for  the 
method  by  orifices. 

259.  The  Flow  of  Water  in  Open  Channels. — For  more 
than  a century  hydraulic  engineers  have  labored  to  find  a fixed 
relation  between  the  slope  and  cross-section  of  a running  stream 
and  the  resulting  mean  velocity.  If  such  a relation  could  be 
found,  then  the  discharge  of  any  stream  could  be  obtained  at 
a minimum  cost.  It  is  now  known  that  there  is  no  such  fixed 
relation.  There  certainly  is  a relation  between  the  bed  of  a 
stream  for  a considerable  distance  above  and  below  the  section. 


* See  Bowie’s  “ Hydraulic  Mining,”  p.  126  (John  Wiley  & Sons,  New  York). 


324 


SUK  VE  YING. 


the  surface  slope,  and  the  resulting  velocity  at  the  section  ; 
but  as  no  two  streams  have  similar  beds,  nor  the  same  stream 
in  different  portions  of  its  length,  and  since  the  bed  character- 
istics are  difficult  to  determine,  and,  furthermore,  are  constantly 
changing  in  channels  in  earth,  the  function  of  bed  cannot  be 
incorporated  into  a formula  to  any  advantage  except  for  chan- 
nels of  strictly  uniform  and  constant  bed,  in  which  case  the 
cross-section  would  sufficiently  indicate  the  bed.  Again,  the 
slope  cannot  be  profitably  introduced  into  a velocity  formula 
except  where  it  is  uniform  for  a considerable  distance  above 
and  below  the  section,  for  the  inertia  of  the  water  tends  to 
produce  uniform  motion  under  varying  slopes,  and  the  effect 
is  that  the  velocity  at  no  point  corresponds  strictly  to  the 
slope  across  that  section.  For  uniform  bed  and  slope,  how- 
ever, formulae  may  be  often  used  to  advantage. 

Let  A — area  of  cross-section  ; 

V = velocity  in  feet  per  second  (=  / for  one  second)  ; 

p =z=  wetted  perimeter  ; 

r — hydraulic  mean  radius  = --  ; 

s — surface-slope  =:  sin  / = ; 

Z = fall  per  length  /; 

Q = quantity  discharged  in  one  second  ; 

S = wetted  surface  in  length  /=://; 

/=  coefficient  of  friction  per  unit  area  of  S; 

p = weight  of  one  cubic  foot  of  water  = density. 


Since  the  friction  varies  directly  as  the  density  and  as  the 
square  of  the  velocity,  we  have  for  the  frictional  resistance  on 
the  mass  covering  the  area  5, 


R = fftSv\ 


(I) 


HYDROGRAPHIC  SURVEYING. 


325 


and  the  work  spent  in  overcoming  this  resistance  in  one  sec- 
ond of  time  is 

K — Rv  — fpSv^ (2) 

If  the  velocity  is  constant,  which  it  is  assunjed  to  be,  then 
this  is  also  the  measure  of  the  work  gravity  does  on  this  mass 


of  water  in  pulling  it  through  the  height  A'  — h"  — Z,  which 
work  is 

K = weight  X fall  — Zf>Q  — ZpvA  ; . . . (3) 

/.  ZpvA  = /pSv\ (4) 

or  (5) 

A Z 

But  S = pi;  - = r;  and  - = sin  z s ; 

P ^ 


T'U  

s or  V ~ c . . c . (6) 


where  c is  an  empirical  coefficient  to  be  determined.  It  is  evi- 
dent that  c is  mostly  a function  of  the  character  of  the  bed, 
and  that  it  can,  therefore,  have  no  fixed  value  for  all  cases. 

Equation  (6)  is  what  is  known  as  the  Chezy  formula.  The 
most  successful  attempt  yet  made  to  give  to  the  coefficient  c 
a value  suitable  to  all  cases  of  constant  flow  is  that  of  Kut- 
ter.'^  Kutter’s  formula,  when  reduced  to  English  foot-units,  is 


* Kutter’s  Hydraulic  Tables,  translated  from  the  German  by  Jackson,  »and 
published  by  Spon,  London,  1876, 


326 


SURVEYING. 


S/rs  = 


. , 1.811  , 0.00281 

41.6  + 

‘ ‘ s 


/ . , 0.0028 1 \ 71 

+ ^41.6+— 


\/rs,  . . (7) 


the  total  coefficient  of  the  radical,  in  brackets,  being  the  eval- 
uation of  c in  equation  (6).  Here  v,  r,  and  s are  the  same 
as  before,  and  ii  is  a “ natural  coefficient  ” dependent  on  the 
nature  of  the  soil,  character  of  bed,  banks,  etc.  Although  it 
was  the  author’s  intention  to  make  a formula  that  would  be 
applicable  even  to  natural  channels,  it  cannot  safely  be  ap- 
plied to  such  unless  they  have  great  uniformity  of  bed  and 
slope. 

The  following  values  of  n are  given  by  Kutter: 


Planed  plank. 

n — 

0.008. 

Pure  cement, 

n = 

.009. 

Sand  and  cement, 

n = 

.010  to  .oil. 

Brickwork  and  ashlar. 

n — 

.012  “ .014. 

Canvas  lining, 

n = 

.015. 

Average  rubble, 

n = 

.017. 

Rammed  gravel. 

n = 

.020. 

In  earth — canals  and  ditches, 

n — 

.020  to  .030, 

depending  on  the  reg- 
ularity of  the  cross- 
section,  freedom  from 
weeds,  etc. 

In  earth  of  irregular  cross-section,  n = .030  to  .040. 

For  torrential  streams,  n = .050. 

In  the  last  two  cases  the  results  are  very  uncertain.  Kut- 
ter’s  tables  are  evaluated  for  7i  = 0.025,  -030,  and  .035. 


HYDROGRAPHIC  SUR  VE  YING. 


327 


The  greatest  objection  to  the  use  of  this  formula  is  the 
labor  involved  in  evaluating  the  “r”  coefficient.  To  facilitate 
the  use  of  the  formula  this  coefficient  has  been  evaluated  for  a 
slope  of  o.ooi  in  Table*  IX.  This  coefficient  changes  so 
slowly  with  a change  in  slope  that  the  error  does  not  exceed 
3|-  per  cent  if  the  table  be  used  for  all  slopes  from  one  in  ten 
to  one  in  5280,  which  is  a foot  in  a mile.  These  tabular  co- 
efficients may  therefore  be  used  in  all  cases  of  ditches,  pipe- 
lines, sewers,  etc.  The  coefficients  are  seen  to  change  rapidly 
for  different  values  of  n,  so  this  value  must  be  chosen  with 
care. 

For  brick  conduits^  such  as  are  used  for  water-supply  and 
for  sewers,  the  formula 


V — i27r°-6=.so-5 

was  found  to  represent  the  experiments  on  the  Boston  con- 
duit, shown  in  Figs.  78  and  79.  This  would  correspond  to  a 
variable  value  of  n in  Kutter’s  formula,  being  nearly  0.012 
however,  as  given  for  brickwork.  This  conduit  is  brick-lined. 
Table  X.*  gives  maximum  discharges  of  such  conduits 
as  computed  by  Kutter’s  formula,  n being  taken  as  0,013. 
The  results  in  heavy  type  include  the  working  part  of  the 
table  for  sewers.  All  less  than  three  feet  per  second  when  the 
depth  of  water  is  one  eighth  of  the  diameter,  or  when  the  flow 
is  one  fiftieth  the  maximum.  This  is  as  small  a velocity  as  is 
consistent  with  a self-cleansing  flow  in  sewers.  All  values 
below  the  heavy-faced  type  correspond  to  velocities  more  than 
fifteen  feet  per  second  when  the  conduit  runs  full,  and  this  is 
as  great  a velocity  as  is  consistent  with  safety  to  the  structure. 
If  the  velocity  is  greater  than  this,  the  conduit  should  be  lined 
with  stone. 

* Taken  from  a paper  by  Robt.  Moore  and  Julius  Baier  in  Journal  of  the 
Association  of  Engineering  Societies^  vol.  v.,  p,  349.  This  table  may  also  be 
used  for  tile  drains. 


328 


SUR  VE  YING. 


The  maximum  flow  does  not  occur  when  the  conduit  runs 
full,  but  when  the  depth  is  about  93  per  cent  of  the  diameter. 
A conduit  or  pipe  will  therefore  not  run  full  except  under 
considerable  pressure  or  head.  The  maximum  velocity  occurs 
when  the  depth  is  about  81  per  cent  of  the  diameter. 

The  relative  mean  velocities  and  discharges  of  a circular 
conduit  for  varying  depths  is  shown  by  the  following  table: 


Depth  of 
Water. 

Relative 

Velocity. 

Relative 

Discharge. 

Depth  of 
Water. 

Relative 

Velocity. 

Relative 

Discharge. 

.1 

.28 

.016 

•7 

.98 

.776 

.2 

.48 

.072 

•75 

•99 

.850 

.25 

•57 

.118 

.8 

•99 

.912 

•3 

.64 

.168 

.81 

1. 00 

.924 

•4 

.76 

.302 

•9 

.98 

.992 

•5 

.86 

•450 

•93 

.96 

1. 000 

.6 

•93 

.620 

1. 00 

.86 

.916 

260.  Cross-sections  of  Least  Resistance. — From  equa- 
tion (6)  of  the  preceding  article  it  is  apparent  that  for  a given 
channel  the  velocity  varies  as  the  square-root  of  the  hydraulic 


mean  radius,  r.  But  r = hence  for  a given  area  of  cross- 

P 

section  the  velocity  is  greater  as  the  wetted  perimeter  is  less. 
The  form  of  cross-section  having  a minimum  perimeter  for  a 
given  area  is  the  circular,  or  for  an  open  channel  the  semicircu- 

7tl^  R 

lar.  In  both  cases  the  hydraulic  mean  radius  is  r = — - = 

where  R is  the  radius  of  the  circle.  Since  it  is  not  always  con- 
venient to  make  the  cross-section  circular  in  the  case  of  ditches 
and  canals,  it  is  evident  that  the  more  nearly  a polygonal 
cross-section  coincides  with  the  circular  form  the  less  will  be 
the  resistance  to  flow.  When  a maximum  flow  is  desired  for 


HYDROGRAPHIC  SUR  VE  YING. 


329 


Fig.  87. 


a given  slope  and  cross-section,  therefore,  the  .shape  should 
conform  as  nearly  as  possible  to  that  of  a semicircle.  To  do 
this,  construct  a semicircle  to  scale  of  the  required  area  of 
cross-section.  Draw  tangents  for  the 
sides  of  the  section  having  the  de- 
sired slope  and  join  these  by  another 
tangent  line  at  bottom,  as  in  Fig. 

87.  This  gives  a little  larger  section- 
al area,  but  some  allowance  should 
be  made  for  accumulations  in  the 

channel.  If  the  slope  is  very  great  and  it  is  desirable  to  re- 
duce the  velocity  of  flow,  it  may  be  done  by  making  the 
channel  wide  and  shallow. 

261.  Sediment-observations. — It  is  often  necessary  in  sur- 
veys of  sediment-bearing  streams  to  determine  the  amount  of 
silt  carried  by  the  water  in  suspension.  The  work  consists  of 
three  operations,  namely:  (i)  obtaining  the  samples  of  water; 
(2)  weighing  or  measuring  out  a specific  portion  of  each,  mix- 
ing these  in  sample  jars  according  to  some  system,  and  setting 
away  to  settle  ; (3)  siphoning  off  the  clear  water,  filtering,  and 
weighing  the  sediment.  Sometimes  a fourth  operation  is  re- 
quired, which  is  to  examine  the  sediment  by  a microscope  on 
a graduated  glass  plate,  and  estimate  the  percentages  of  differ- 
ent-sized grains.  The  sedimentary  matter  carried  in  suspen- 
sion may  be  divided  into  two  general  classes, — that  in  continu- 
ous suspension,  and  that  in  discontinuous  suspension.  The 
former  is  composed  of  very  fine  particles  of  clay  and  mud 
whose  specific  gravity  is  about  unity,  so  that  any  slight  dis- 
turbance of  the  water  will  prevent  its  deposition.  This  once 
taken  up  by  a running  stream  is  carried  to  its  mouth  or  caught 
in  stagnant  places  by  the  way.  The  matter  in  discontinuous 
suspension  consists  of  sand,  more  or  less  fine  according  to  the 
velocity  and  agitation  of  the  current.  This  matter  is  con- 
stantly falling  towards  the  bottom  and  is  only  prevented  by  the 


330 


SURVEYING. 


violent  motions  of  the  medium  in  which  they  are  suspended. 
These  particles  are  constantly  being  picked  up  where  the  ve- 
locity is  greater,  and  dropped  again  where  the  velocity  is  less. 
A natural  channel  will  therefore  carry  about  the  same  per- 
centage of  fine  or  continuous  matter  between 
two  consecutive  tributaries,  but  of  the  coarser 
material  there  will  be  no  uniformity  whatever  in 
successive  sections  in  this  same  stretch  of  river. 
In  natural  channels  there  are  always  alternate 
engorged  and  enlarged  sections  for  any  particu- 
lar stage  of  river,  and  the  positions  of  these  en- 
gorgements and  enlargements  are  different  for 
different  stages.  In  fact,  the  engorged  sections 
at  high  water  are  usually  the  enlarged  sections 
at  low  water,  and  vice  versa.  If  the  bed  is  fria- 
ble the  engorged  section  is  always  enlarging,  and 
the  enlarged  section  is  constantly  filling  as  a 
result  of  the  discontinuous  movement  of  sedi- 
mentary matter.  The  cause  of  these  relative 
changes  of  position  of  engorged  and  enlarged 
sections  is  the  great  variation  in  width. 

It  is  the  discontinuous  sediment  which  is  of 
principal  significance  to  the  engineer,  for  this 
is  the  material  from  which  sand-bars  are  formed 
which  obstruct  navigation,  and  it  is  also  the  ma- 
terial from  which  he  builds  his  great  contraction 
works  behind  his  permeable  dikes.  The  water 
being  partially  checked  behind  these  dikes  at  once  drops  the 
heavier  sediment,  and  so  artificial  banks  are  rapidly  formed. 
The  continuous  sediment  is  of  little  consequence  to  the  engi- 
neer. 


* See  paper  by  the  author  entitled  “ Three  Problems  in  River  Physics,”  be- 
fore the  American  Association  for  the  Advancement  of  Science,  Philadelphia 
meeting,  1884. 


H YDROGRAPHIC  SUR  VE  YING. 


331 


262,  Collecting  the  Specimens  of  Water. — It  is  neces- 
sary to  take  samples  of  water  from  various  parts  of  the  cross- 
section  in  order  to  obtain  a fair  average.  Surface  and  bottom 
specimens  should  always  be  taken,  and  if  great  exactness  is 
required  specimens  should  also  be  taken  at  mid-depth.  One 
of  each  of  these  should  be  taken  at  two  or  three  points  on  the 
cross-section.  A full  set  of  specimens  is  collected  once  or 
twice  a day.  A special  apparatus  is  required  for  obtaining 
samples  from  points  beneath  the  surface.  The  requirements 
of  such  an  apparatus  are  very  well  satisfied  by  the  device 
shown  in  Fig.  88,  which  the  author  designed  and  used  very 
successfully  in  a hydrographic  survey  of  the  Mississippi  River 
at  Helena,  Ark.,  in  1879.^  C is  a.  galvanized  iron  or  copper 
cup  ; /an  iron  bar  one  inch  square;  L a mass  of  lead  moulded 
on  the  bar  at  bottom ; B the  bottom  cup  for  bringing  to 
the  surface  a specimen  of  the  bottom,  I being  a leather  cover; 
W the  springing  wire  by  which  the  lids  a a are  released  and 
drawn  together  by  the  rubber  bands  b b when  the  apparatus 
strikes  the  bottom,  or  when  this  wire  is  pulled  by  an  auxil- 
iary cord  from  above ; d d adjustable  hinges  allowing  a tight 
joint  on  the  rubber  packing-disks  c c when  the  lids  are  closed. 
In  descending,  the  lids  are  open  and  the  water  in  the  can  C is 
always  a fair  sample  of  the  water  surrounding  the  apparatus. 
When  the  lids  are  closed  the  sample  is  brought  securely  to 
the  surface.  The  can  when  closed  should  be  practically  water- 
tight ; if  it  leaks  at  bottom  some  of  the  heavier  sediment  is 
likely  to  escape,  for  it  settles  very  quickly.  The  bottom  speci- 
men should  be  taken  about  a foot  above  the  bottom  to  avoid 
getting  an  undue  portion  of  sand  which  is  at  once  stirred  up 
by  the  apparatus  striking  the  bottom. 

263.  Measuring  out  the  Samples. — A given  portion  of  each 
specimen  by  measure  or  by  weight  is  selected  for  deposition. 


* See  Report  of  Chief  of  Engrs.,  U.  S.  A.,  1879,  vol.  iii.,  p,  1963. 


332 


SURVEYING. 


Great  care  must  be  exercised  in  obtaining  the  sample  volume. 
It  cannot  be  poured  off,  even  after  violent  shaking,  for  the 
heavy  sand  falls  rapidly  to  the  bottom.  A good  way  is  to 
draw  it  from  the  vessel  by  an  aperture  in  its  side  while  the 
water  is  stirred  within  ; greater  accuracy  can  be  attained  by 
weighing  the  sample  of  water  than  by  measuring  it.  All  the 
samples  of  a given  kind  are  then  put  together  in  one  jar,  which 
is  properly  labelled,  and  set  away  to  settle.  Thus,  all  the  sur- 
face samples  are  put  into  one  jar,  the  mid-depth  samples  in 
another.  The  Mississippi  and  the  Missouri  River  water  re- 
quires about  ten  days’  settling  to  become  clear. 

264.  Siphoning  off,  Filtering,  and  Weighing  the  Sedi- 
ment.— When  the  water  has  become  quite  clear  it  is  carefully 
siphoned  off,  and  the  residue  is  filtered  through  fine  filter  paper 
(Munktell’s  is  best).  Two  papers  are  cut  and  made  of  exactly 
the  same  weight.  One  is  used  for  filtering  and  the  duplicate 
laid  aside.  The  filter-paper  containing  the  sediment  and  also 
its  duplicate  are  then  dried  in  an  oven  at  a temperature  not 
higher  than  180*^.  When  quite  dry  the  sediment  paper  is  put 
in  one  pan  of  the  balance,  and  the  duplicate  in  the  other  and 
weights  added  to  balance.  The  sum  of  the  weights  is  the 
weight  of  the  sediment.  This  divided  by  the  weight  of  the 
sample  of  water,  usually  expressed  by  a vulgar  fraction  whose 
numerator  is  one,  is  the  proportionate  quantity  sought. 


CHAPTER  XI. 


MINING  SURVEYING. 

265.  Definitions. — Mining  Surveying,  like  all  other  classes 
of  surveying,  has  for  its  object  the  determination  pf  the  rela- 
tive positions  of  the  different  portions  of  the  subject  of  the 
survey.  The  same  principles  which  are  employed  in  surveying 
on  the  surface  govern  the  engineer  in  the  prosecution  of  a 
mining  survey.  In  fact,  mining  surveying  may  be  considered 
as  an  extension  of  topographical  surveying  to  the  accessible 
portions  beneath  the  surface  of  the  earth,  with  certain  modi- 
fications of  the  adjuncts  of  surface  surveying,  necessitated  by 
the  nature  of  the  case. 

The  parts  of  a mine  included  in  a mining  survey  are  the 
surface  and  surface-workings,  shafts,  tunnels,  inclines  or  slopes, 
drifts,  stopes,  winzes,  cross-cuts,  levels,  air-courses,  entries,  and 
chambers. 

Surface-workings  include  open  cuts,  pits,  and  other  exca- 
vations of  limited  extent. 

A Shaft  is  a pit  sunk  from  the  surface  more  or  less  perpen- 
dicularly on  the  vein  or  to  cut  the  vein.  The  inclination  of 
the  vein  is  called  the  Dip,  or  Pitch,  and  its  direction  across  the 
country  is  called  the  Strike. 

A Tunnel  is  a horizontal  excavation  from  the  surface  along 
the  course  of  a vein,  or  across  the  course  of  known  veins,  for 
purposes  of  discovery.  The  approach  to  a tunnel  is  called  an 
Adit. 

An  Incline  or  Slope  is  a tunnel  run  at  an  angle  to  the  hori- 
zontal. 


334 


SURVEYING. 


A Drift  is  a tunnel  starting  from  an  underground  working 
such  as  a shaft.  When  there  are  a series  of  drifts  at  different 
depths,  they  are  termed  Levels;  as  first  level,  second  level, 
or  50-foot  level,  lOO-foot  level,  etc. 

A Stope  is  the  working  above  or  below  a level  from  which 
the  ore  is  extracted.  An  overhand  or  back  stope  is  the  work- 
ing above  a level ; an  underhand  stope  is  the  working  from 
the  floor. 


A Winze  is  a shaft  sunk  from  a level. 

A Cross-cut  is  a level  driven  across  the  course  of  a vein. 

An  Air-course  is  a tunnel  driven  for  the  purpose  of  venti- 
lation. 

An  Entry  is  a passage  through  a mine. 

A Chamber  is  a large  room  from  which  the  ore  is  mined. 
The  last  two  terms  are  used  more  especially  in  coal-mines, 
where  the  vein  lies  flat  or  nearly  so. 


MINING  SURVEYING. 


335 


The  operations  of  a mining  survey  are  conducted  like  those 
of  a topographical  survey.  An  initial  point  is  selected  usually 
from  its  importance  to  the  object  sought,  and  all  the  subse- 
quent stations  are  connected  either  directly  or  indirectly  with 
it,  and  their  positions  with  reference  to  it  shown  on  the  map 
of  the  survey. 

266.  Stations  are  occupied  by  candles  or  lamps  constructed 
for  the  purpose,  in  place  of  poles,  flags,  etc.,  as  on  the  surface. 
An  illuminated  plumb-line  is  a good  substitute  for  a lamp,  and 
gives  the  observer  a greater  vertical  range,  which  is  helpful  in 
case  the  station  is  obscured  by  intervening  objects. 

Owing  to  the  peculiar  nature  of  the  survey,  it  is  imprac- 
ticable and  sometimes  inexpedient  to  mark  stations  as  on  the 
surface  ; recourse  is  therefore  had  to  other  devices,  which  must 
be  employed  to  suit  circumstances. 

It  would  not  be  advisable,  even  if  it  were  practicable,  to 
leave  a station-mark  on  the  bottom  of  any  portion  of  a mine, 
as  frequent  passing  would  disturb  or  obliterate  it. 

It  is  better  therefore  to  leave  the  mark  overhead,  if  acces- 
sible and  not  liable  to  be  disturbed,  either  by  driving  a nail  in 
a timber  should  one  be  convenient,  or  by  drilling  a hole  in 
which  may  be  inserted  a wooden  plug,  properly  marked,  or 
simply  by  cutting  a cross  or  other  device  on  the  exposed  sur- 
face. Another  method,  where  the  above  cannot  be  employed, 
is  by  marking  points  on  the  walls  and  measuring  the  respective 
distances  from  the  station  to  them. 

267.  Instruments. — Steel  tapes  only  should  be  used  for 
measuring,  being  more  convenient  and  less  liable  to  inaccuracy 
than  a chain.  These  may  be  of  different  lengths  to  suit  the 
work  on  which  they  are  to  be  employed  ; sometimes,  as  in  the 
case  of  coal-mine  surveys,  tapes  of  several  hundred  feet  in 
length  can  be  employed  to  advantage. 

The  Compass.,  unless  used  as  an  angular  instrument  to  de- 
flect from  an  established  line,  should  not  be  employed  in  min- 


33*5 


SUK  VE  YING. 


ing  surveys,  as  the  variation  between  stations  is  so  inconstant 
as  to  render  it  unreliable  when  used  to  deflect  from  the  mag- 
netic meridian.  The  magnetic  needle  may  be  used,  however, 
in  connection  with  the  transit  as  a check. 

The  Transit  alone  should  be  used  in  important  work,  and 
certain  additions  to  it  for  vertical  pointings  will  be  found  indis- 
pensable. In  sighting  up  or  down  a shaft  the 
ordinary  form  becomes  r.seless  when  the  line 
of  sight  passes  inside  the  upper  plate  of  the 
instrument.  A prismatic  eye-piece.  Fig.  90, 
will  overcome  this  difficulty  for  upward  sights, 
but  the  survey  cannot  be  carried  downward  by 
its  use> 

An  extra  telescope  attached  either  to  the  top  or  side  of  the 
central  telescope  will  overcome  this  difficulty.  The  attach- 
ment to  the  top  is  made  as  shown  in  Fig.  91  by  coupling-nuts, 


Fig,  91. 

which  fasten  it  firmly  over  the  centre  of  the  instrument.  The 
attachment  to  the  side,  Fig.  92,  is  effected  by  means  of  a spindle 
from  the  attached  telescope  which  fits  into  the  hollow  axis  of 
the  central  telescope  and  is  secured  by  means  of  a clip  which 


MINING  SURVEYING. 


337 


passes  through  both  the  axis  and  spindle.  A counterpoise  is 
similarly  attached  to  the  opposite  side  of  the  central  telescope 
to  preserve  the  equilibrium. 

This  latter  form  of  attachment  is  more  com- 
pact than  the  former,  the  principal  objection  to 
it  being  that  a correction  must  be  applied  to 
each  reading  of  a horizontal  angle  equal  to  the 
tangent  of  an  angle  whose  opposite  side  is  the 
distance  between  the  centres  of  the  telescopes, 
and  whose  adjacent  side  is  the  horizontal  dis- 
tance between  stations.  This  objection,  how- 


Fig.  92. 


Fig,  93. 


ever,  is  removed  by  a simple  device.  Two  brass  tubes,  Fig.  93, 
about  two  inches  long,  are  connected  by  an  intermediate  web, 
so  that  the  distance  between  their  centres  shall  exactly  equal 
that  between  the  centres  of  the  telescopes.  One  of  the  tubes 
is  of  sufficient  size  to  enclose  a pike-staff  graduated  to  fractions 
of  a foot,  upon  which  it  can  be  easily  moved  to  any  desired 
22 


338 


SUA'  VE  Y/NG. 


height,  and  the  other  large  enougli  to  contain  a candle,  and  has 
a light  plumb-bob  suspended  below  its  centre  the  better  to 
maintain  the  staff  in  a perpendicular  position  ; the  staff  now 
being  placed  over  any  station  and  the  brass  tube  and  candle 
set  at  the  height  of  the  instrument  on  the  staff,  which  is  held 
in  a perpendicular  position  with  the  line  between  the  tubes 
parallel  to  the  horizontal  axis  of  the  telescope,  a reading  can 
be  made  to  the  flame  of  the  candle  which  gives  at  once  the 
true  azimuth  of  the  line  and  the  dip  of  the  shaft. 

In  using  this  device  the  side  telescope  and  tube  carrying 
the  candle  should  always  be  on  the  same  side  of  the  line.  The 
transit  must  always  be  placed  exactly  over  the  point  occupied 
by  the  foot  of  the  staff ; and  here  it  may  be  well  to  state  that 
the  greatest  care  and  accuracy  must  be  exercised  in  exactly 
centring  the  instrument  over  the  station,  as  the  courses  are 
carried  forward  entirely  by  deflection  angles,  so  that  an  error 
introduced  at  one  station  is  carried  through  all  and  increased 
at  each.  Again,  in  sighting  down  a shaft,  although  the  per- 
pendicular distance  may  be  considerable,  the  horizontal  dis- 
tance between  stations  must  be  small,  so  that  even  a slight 
error  made  in  a shaft  will  be  of  considerable  magnitude  when 
carried  out  in  the  levels  of  a large  mine. 

The  transit  should  have  an  extension  tripod.  Fig.  94,  so 
that  one  or  more  of  the  legs  can  be  shortened,  the  better  to 
place  it  over  a station  on  a steep  mountain-side  or  in  a mine, 
or  to  lower  the  instrument  to  see  under  intervening  objects,  or 
to  adapt  it  to  different  heights  of  the  workings  of  the  mine. 

The  Mining  Transit  should  be  provided  with  the  Solar  At- 
tachment, that  all  lines  of  the  survey  may  be  referred  to  the 
meridian. 

In  unimportant  surveys  the  pitch  of  the  shaft  or  the  dip  of 
the  vein  may  be  determined  by  the  clinometer  or  by  measuring 
the  horizontal  and  perpendicular  distances  between  any  two 
conveniently  located  points  of  the  foot  or  hanging  wall  and 


MINING  SURVEYING. 


339 


calculating  the  pitch  or  dip  from  the  measurements  thus  ob- 
tained. 

A plummet-lamp,  Fig.  95,  will  also  be  found  very  convenient. 


Fig.  94.  Fig.  95. 


268.  Mining  Claims. — The  first  work  of  the  surveyor  upon 
a mining  claim  is  its  location.  Mining  claims  are  of  different 
dimensions  according  to  the  local  laws  and  customs  of  the 


340 


SUR  VE  YING. 


country  ; varying  from  50  feet  to  600  feet  in  width  and  from 
100  to  3000  feet  in  length.  In  the  earliest  days  of  Western 
mining,  the  dimensions  of  a claim  were  decided  at  a convention 
of  all  the  miners  in  the  district.  Now  the  United  States  laws 
limit  the  length  to  1500  feet,  but  the  width  still  varies  not 
only  in  different  States,  but  in  different  counties  in  the  same 
State. 

The  form  of  a mining  claim  is  essentially  a parallelogram, 
being  regulated  by  the  U.  S.  mining  laws,  which  prescribe  that 
whatever  the  relative  position  of  the  side  lines  to  each  other, 
the  end  lines  must  be  parallel. 

This  is  to  prevent  more  than  fifteen  hundred  feet  of  a vein 
or  lode  from  being  included  in  one  claim.  The  side  lines  of  a 
claim  may  be  straight  lines  extending  between  the  ends  of  op- 
posite end  lines,  or  they  may  be  broken  lines  to  include  the 
vein  if  it  should  be  curved,  so  as  to  pass  outside  straight  lines  ; 
but  in  any  case  they  can  only  include  1500  feet  of  the  vein 
measured  along  the  centre  line  of  the  claim.  A mining  claim 
is  included  between  parallel  vertical  planes  passed  through  the 
end  lines ; but  a miner  has  a right  to  follow  his  vein  downward, 
although  it  so  far  passes  from  the  perpendicular  in  its  down- 
ward course  as  to  extend  beyond  vertical  planes  passed 
through  its  side  lines. 

The  above  are  the  essential  features  which  govern  the  shape 
of  a mining  claim. 

The  method  of  procedure  in  making  the  location  is  as  fol- 
lows: When  the  discoverer  of  a mine  has  sunk  his  shaft  ac- 
cording to  law,  so  as  to  expose  10  feet  of  the  vein,  he  is  entitled 
to  have  his  claim  surveyed  and  recorded.  He  then  decides 
how  much  of  the  1 500  feet  he  desires  to  extend  on  either  side  of 
his  discovery-shaft  along  the  vein.  He  is  governed  in  this 
by  various  considerations,  such  as  his  proximity  to  other  claims, 
the  promise  of  mineral  in  different  portions  of  the  lode,  or  the 
nature  of  the  ground.  The  surveyor  begins  his  survey  for 


Fig.  96. 


342 


SUI^  VE  YING. 


location  at  the  point  of  discovery,  and  runs  from  it  in  opposite 
directions  until  he  has  measured  off  15CX)  feet. 

The  survey  has  thus  far  been  run  on  the  centre  line  of  the 
claim.  On  arriving  at  the  ends  of  the  line,  the  surveyor 
measures  off  half  the  width  of  the  claim  on  each  side  of  the 
centre  line,  generally  at  right  angles  to  it  if  the  claim  is 
straight,  and  sets  his  corners  at  the  ends  of  the  end  lines.  He 
also  places  monuments  on  the  side  lines,  midway  between  the 
corners,  called  the  side-line  centre  monuments,  the  law  re- 
quiring that  the  claim  shall  be  distinctly  marked  upon  the 
ground,  so  that  its  boundaries  can  be  readily  traced.  This 
much  of  the  survey  being  now  completed,  it  remains  to  run  a 
tie  line  from  some  corner  of  the  claim  to  a well-known  monu- 
ment. This  must  be  a section  corner  of  the  Government 
surveys  if  the  claim  be  on  surveyed  lands,  otherwise  to  a 
prominent  natural  object,  or  to  a locating  monument  estab- 
lished for  the  purpose.  This  is  done  to  identify  and  locate 
the  claim  so  that  its  locus  may  be  a matter  of  record.  An 
example  of  an  actual  plat  of  mining  claims  is  shown  in  Fig.  96. 
The  next  survey  of  a mining  claim  is  its  survey  for  patent  of 
the  United  States.  The  original  location  maybe  made  by 
any  surveyor,  and  is  sometimes  made  by  the  miner  himself; 
but  the  survey  on  which  the  patent  or  title  from  the  United 
States  is  issued  must  be  made  by  a deputy  of  the  U.  S. 
Surveyor-general  of  the.  Public  Lands,  who  is  thus  known  as  a 
U.  S.  Deputy-mineral-surveyor.  These  officers  give  bonds  to 
the  Government  in  the  sum  of  $10,000  for  the  faithful  perform- 
ance of  their  work,  and  are  required  to  pass  an  examination, 
that  the  Surveyor-general  may  be  satisfied  of  their  capability. 

The  survey  for  patent  must  be  made  with  the  greatest  care 
and  accuracy.  It  must  exactly  locate  the  claim  with  reference 
to  a corner  of  the  public  surveys,  if  such  be  within  two  miles, 
and  must  show  the  nature  and  extent  of  the  conflict  with 
other  official  surveys  if  it  should  conflict  with  any,  or  with 


MINING  SURVEYING, 


343 


other  mining  claims  not  officially  surveyed  if  it  is  desired  to 
exclude  from  the  claim  the  area  in  conflict. 

A specimen  of  the  field-notes  of  a survey  for  patent  issued 
for  the  instruction  of  the  U.  S.  Deputy-surveyors  of  Colorado 
is  given  in  Appendix  B. 

The  Surveyors-general  of  the  different  States  and  Territo- 
ries issue  instructions  to  their  deputies,  and  these,  with  a knowl- 
edge of  the  U.  S.  mining  laws,  must  govern  the  surveyor  in  his 
work;  but  as  they  are  more  strictly  legal  than  mathematical,  it 
is  not  important  to  consider  them  in  this  chapter.* 

The  foregoing  surveys  are  strictly  land  surveys,  and  are 
only  mentioned  to  illustrate  the  method  of  staking  out  a min- 
ing claim  and  to  give  some  idea  of  the  shape  and  size. 

UNDERGROUND  SURVEYS. 

269.  Mining  Surveying  proper,  or  the  underground  work 
of  the  survey,  will  be  considered  in  a few  practical  examples 
selected  from  actual  cases. 

270.  To  determine  the  Position  of  the  End  or  Breast  of 
a Tunnel  and  its  Depth  below  the  Surface  at  that  Point. — 

Set  up  the  instrument  at  a point  outside  the  tunnel,  so  as  to 
command  as  long  a sight  as  possible  into  the  tunnel  and  also 
the  surface  of  the  mountain  above  it.  If  the  end  of  the  tun- 
nel can  be  seen  from  the  station  a course  and  distance  can  be 
taken  at  once  to  the  breast,  and  this  course  and  distance  dupli- 
cated on  the  surface.  Vertical  angles  can  then  be  measured 
to  the  points  thus  determined  on  the  surface  and  in  the  tun- 
nel, and  the  calculation  of  the  depth  of  the  breast  below  the 
surface  may  be  made  from  the  data  thus  obtained. 

* Copies  of  the  Instructions  can  be  procured  of  any  Surveyor-general  on 
application.  Those  for  Colorado  are  given  in  Appendix  B.  The  U.  S.  mining 
laws,  together  with  all  State  and  Territorial  laws  and  local  mining  rules  and 
regulations,  are  compiled  in  vol.  xiv.  of  the  U.  S.  Census  for  1880,  4to,  705  pp., 
1885.  This  is  a most  valuable  publication.  Price  $4  if  not  obtained  through 
an  M.  C. 


344 


SURVEYING. 


In  case  the  breast  is  not  visible  from  the  first  station,  take 
as  long  a sight  as  practicable  to  Station  No.  2,  and  before  re- 
moving the  instrument  reproduce  Station  No.  2 upon  the  sur- 


Fig.  97. 

face  as  in  the  preceding  case,  thus  avoiding  a resetting  of  the 
instrument  at  Station  No.  i when  the  underground  work  is 
completed.  At  the  same  time  measure  the  vertical  angle  to 
Station  No.  2 in  the  tunnel.  Set  the  instrument  at  Station 
No.  2,  and,  after  having  obtained  the  back  readings  to  Station 
No.  I,  measure  the  course,  distance,  and  vertical  angle  to  Sta- 
tion No.  3. 

Repeat  the  above  operations  at  the  different  stations  until 


MINING  SUR  VE  YING.  . 


345 


the  breast  is  reached,  taking  any  measurements  of  the  dimen- 
sions of  the  work  that  may  be  necessary,  and  leaving  station 
marks  for  future  reference,  as  described  in  article  266.  Set  the 
instrument  over  Station  No.  2 on  the  surface  and  very  care- 
fully duplicate  the  courses  and  distances  measured  in  the  tun- 
nel, at  the  same  time  noting  the  vertical  angles  between  the 
surface  stations.  The  vertical  angles  can  be  measured  most 
easily  by  sighting -to  a point  on  a short  staff  at  a height  above 
the  station  equal  to  the  height  of  the  instrument. 

It  is  advisable  to  explore  the  tunnel  before  surveying  it,  as 
then  any  difficulties  can  be  provided  for  and  the  stations 
selected  more  advantageously.  Sometimes  the  course  from 
Station  No.  I to  Station  No.  2 is  assumed  as  a meridian  of  the 
survey  and  all  courses  deflected  from  it,  but  it  is  better  to  use 
the  true  solar  course  between  these  stations  because  the  field 
notes  can  then  be  placed  in  the  table  for  calculation  without 
further  reduction. 


Example. — Following  is  a specimen  of  field-notes  of  the  survey  of  a tunnel 
both  underground  and  surface: 

FIELD-NOTES. 


Station. 

Verticai 

in  Tunnel. 

L Angles 

on 

Surface. 

Course. 

Distance. 

Remarks. 

S.  36°  50'  W. 
S.  36°  50'  W. 

19. 1 ft. 

99.1 

to  mouth  of  tunnel. 

4-  1°  18' 

4-10°  35' 

to  Sta.  No.  2. 

2 

4-  0°  31' 

+15“  43' 

S.  49°  47'  W. 

104.2 

“ “ 3. 

3 

4-  0°  45' 

4-14°  27' 

S.  40°  0'  W. 

37.1 

“ “ 4. 

4 

1 

0 

3 

4-16°  17' 

S.  4°55'E. 

56.5 

“ “ 5. 

5 

4-  3°  37' 

4-12“  21' 

S.  71“  15'  E. 

46.0 

“ “ 6. 

6 

4-  3°  30' 

4-13°  56' 

S.  77°  30'  E. 

40.7 

to  breast  of  tunnel. 

Breast.. 

-17'’  56' 

N.  16°  16'  E. 

266.57 

from  station  on  surface 

over  breast  of  tunnel  to 

Sta.  No.  I. 

346 


SURVEYING. 


The  following  table  shows  the  method  of  reducing  the  survey.  The  first 
six  columns  represent  the  ordinary  method  of  reducing  a traverse  to  a straight 
line.  The  agreement  between  the  resultant  and  the  check  course  proves  the 
accuracy  of  the  field  work. 

Columns  7 and  8 contain  the  vertical  angles  in  the  tunnel  and  the  rise  or 
fall  in  feet  corresponding  to  them. 

Columns  9 and  10  similarly  contain  the  vertical  angles  of  the  courses  and 
distances  on  the  surface  and  the  difference  of  elevations  between  stations  corre- 
sponding to  them. 

The  algebraic  sum  of  the  vertical  heights  in  the  tunnel  gives  the  difference 
of  level  between  the  Station  No.  i and  the  breast  ; and  the  sum  of  the  differ- 
ences of  elevation  in  column  10  gives  the  total  difference  of  elevation  between 
Station  No.  i and  the  point  on  the  surface  over  the  breast. 

The  difference  of  columns  8 and  10  shows  the  depth  of  the  breast  of  the 
tunnel  below  the  surface.  The  tangent  of  the  angle  obtained  by  dividing  the 
sum  of  the  elevations  in  column  10  by  the  length  of  the  resultant  distance 
should  agree  with  the  vertical  angle  read  to  Station  No.  i from  the  point  on 
the  surface  over  the  breast  of  the  tunnel. 

The  following  is  the  form  used  in  reducing  the  field-notes: 


OFFICE  FORM. 


Course. 

Dist. 

Latitude. 

Departure. 

Vert. 

Angle. 

Rise  or 
Fall. 

Vert. 

Angle. 

Rise  or 
Fall. 

N. 

S. 

E. 

W. 

S.  36°  50'  W. 

99.1 

79-32 

59-41 

+ 1°  18' 

+ 2-24 

+ 10®  35' 

4-  18.51 

S.  49°  4/  w. 

104.2 

67.28 

79-57 

+ 0°  31' 

+ 0-94 

+ 15“  43' 

+ 29.32 

S.  40°  00'  w. 

37-1 

28.42 

23.84 

+ 0°  45' 

+ 0.48 

+ 14°  27' 

+ 9-56 

S.  4°  55'  E. 

56.5 

56.29 

4.84 

- 0°  34' 

— 0.56 

+ 16°  17' 

+ 16.50 

S.  71°  15'  E. 

46 

14.78 

43-56 

+ 3“  37' 

+ 2.91 

4-  12®  21' 

+ 10.07 

S.  77°  30'  E. 

40.7 

8.81 

39-73 

+ 3°  30' 

+ 2.49 

+ 13“  56' 

4-  10. 10 

Resultant  course, 

N.  16®  20'  E. 

265.62 

254.90 

74-69 

Total  -{-  8.50 

Total  -f-  94.06 

254.90  254.90  162.82  162.82  8.50 

Depth  below  surface  = 85.56 

Check : 85.56  ■+■  265.62  = 0.3221  = tan  17°  51'. 


271.  Required,  the  Distance  that  a Tunnel  will  have  to 
be  driven  to  cut  a Vein  with  a Certain  Dip. — Case  I.  W/ien 
the  direction  of  the  tunnel  is  at  right  angles  to  the  course,  or  par- 
allel to  the  pitch  of  the  vem.  The  dip  having  been  first  ascer- 


MINING  SURVEYING. 


347 


tained  by  sighting  down  a shaft  sunk  on  the  vein,  or  by  any 
other  practicable  method,  set  up  the  instrument  on  the  apex 
or  outcrop  of  the  vein  directly  over  the  line  of  the  proposed 
tunnel  and  measure  the  vertical  angle  and  horizontal  distance 
to  the  mouth  of  the  tunnel. 

From  the  results  obtained  calculate  the  depth  at  which  the 
tunnel  will  intersect  the  vein,  then  from  this  depth  and  the 
angle  of  the  dip,  calculate  the  horizontal  distance  of  the  vein 
from  a vertical  line  through  the  instrument  station. 

This  distance,  added  to  or  subtracted  from  the  horizontal 
distance  between  the  station  and  mouth  of  the  tunnel,  accord- 
ing as  the  dip  is  from  or  toward  the  mouth,  will  give  the  re- 
quired distance. 


Example. — A certain  vein  has  a course  of  N.  45°  E.  and  its  pitch  is  N. 
45°  W.  with  a dip  of  is”  from  a vertical  line.  The  horizontal  distance  to  the 
mouth  of  a cross  cut  tunnel  from  the  apex  of  the  vein  is  200  feet  S.  45”  E.  and 
the  vertical  angle  is  — 25“. 

At  what  distance  from  the  mouth  will  the  tunnel  intersect  the  vein?  (Fig.  98.) 


348 


SURVEYING. 


Depth  at  which  the  tunnel  will  intersect  the  vein  = 200  X tan.  25®  = 93.26 
ft. 

Distance  of  vein  from  vertical  line  at  depth  of  93,26  ft.  = 93.26  X tan.  15“ 

= 24.99 

Adding  the  last  result  to  the  horizontal  measured  distance,  we  have  24.99 
-j-  200  = 224.99  ^he  distance  from  the  mouth  of  the  tunnel  to  its  intersection 
with  the  vein. 

Case  II.  IV/icn  the  direction  of  the  tunnel  is  oblique  to  the 
course  of  the  vein.  Proceed  as  in  Case  I.  and  measure  the  hori- 
zontal distance  from  the  instrument  station  to  the  mouth  of 
the  tunnel,  the  vertical  angle  and  the  dip,  also  the  angle  which 
the  course  of  the  vein  and  the  line  of  the  tunnel  make  with 
each  other.  Calculate  the  depth  at  which  the  tunnel  will  in- 
tersect the  vein,  and  the  distance  of  the  vein  at  the  tunnel  level 
from  a vertical  line  through  the  station,  as  in  the  previous  case. 
Multiply  this  distance  by  the  cosectant  of  the  angle  between 
the  courses  of  the  vein  and  tunnel  and  apply  it  to  the  meas- 
ured horizontal  distance,  as  in  Case  I.,  and  we  have  the  re- 
quired result. 

Example, — A certain  vein  has  a course  of  N.  45®  E.  and  its  pitch  is  N.  45® 
W.  with  a dip  of  15°  from  a vertical  line.  The  horizontal  distance  to  the 
mouth  of  a cross-cut  tunnel  running  due  west  is  200  ft.  due  east  and  the 
vertical  angle  is  —25°. 

At  what  distance  from  the  mouth  will  the  tunnel  intersect  the  vein  ? (Fig.  99.) 

Depth  at  which  the  tunnel  will  intersect  the  vein  = 200  X tan,  25°  = 93.26  ft. 

Distance  of  vein  from  vertical  line  at  depth  of  93.26  ft,  = 24.99 

Angle  between  course  of  vein  and  line  of  tunnel  = 45°. 

Multiplying,  24.99  X cosec.  45°  = 35.34  ft.  Add  the  result  to  the  horizontal 
measured  distance  and  we  have  2004-35.34  = 235.34  ft.,  the  required  distance 
from  the  mouth  of  the  tunnel  to  its  intersection  with  the  vein. 


272.  Required,  the  Direction  and  Distance  from  the 
Breast  of  a Tunnel  to  a Shaft,  and  the  Depth  at  which  it 
will  cut  the  Shaft. — Make  a survey  of  the  tunnel  and  repro- 
duce it  upon  the  surface,  as  in  the  first  example.  Calculate 
the  depth  of  the  breast  below  the  surface.  Set  up  the  instru- 


MINING  SURVEYING. 


349 


merit  at  the  shaft  and  measure  the  vertical  angle  and  hori- 
zontal distance  to  the  point  on  the  surface  over  the  breast. 
Calculate  their  difference  of  level  from  the  measurements  ob- 


tained and  add  it  to  or  subtract  it  from  the  depth  of  the 
breast  below  the  surface.  The  result  is  the  depth  of  tunnel 
below  the  mouth  of  the  shaft. 

Survey  the  shaft  to  a point  whose  vertical  depth  is  equal 


350 


SUK  VE  YING. 


to  the  depth  of  the  tunnel  level.  Calculate  the  horizontal 
distance  and  direction  of  this  point  from  the  instrument 
station  at  the  mouth  of  the  shaft,  and  mark  its  position  upon 
the  surface.  Connect  it  with  the  point  marking  the  position 
of  the  breast  of  the  tunnel,  and  we  have  the  line  required. 
From  the  information  thus  obtained  range-lines  can  be  sus- 
pended in  the  tunnel,  to  give  the  direction  of  the  shaft  from 
the  breast. 

Example. — A shaft  whose  centre  at  the  surface  bears  S.  65“  E.  73  ft.,  verti- 
cal angle  10°  20',  from  a point  on  the  surface  over  the  breast  of  the  tunnel 
in  the  first  example,  has  a pitch  of  14°  30'  N.  2°  15'  W.  from  a vertical  line. 
At  what  direction  and  distance  from  the  breast  of  the  tunnel  will  it  cut  the 
shaft,  and  at  what  depth  ? (Fig.  97.) 

The  following  is  the  form  of  the  field-notes: 


FIELD-NO'TES. 


Station, 

Vertical  Angles. 

Course. 

Distance. 

•Remarks. 

In 

Tunnel. 

On 

Surface. 

I 

S.  36°  50'  W. 

19. 1 

To  mouth  of  tunnel. 

+ 1°  i8' 

+ io»  35' 

S.  36°  50'  W. 

99.1 

“ Station  No.  2. 

2 

-f  0°  31' 

+ 15°  43' 

S,  49°  47'  W. 

104.2 

“ “ “ 3. 

3 

+ 0°  45' 

+ 14°  27' 

S.  40°  00'  W. 

37-1 

4. 

4 

- 0°  34' 

4- 16°  17' 

S.  4°  55'  E. 

56.5 

5 

+ 3°  37' 

4- 12°  21' 

S.  71“  15'  E. 

46.0 

“ “ “ 6. 

6 

+ 3°  30' 

4- 13°  56' 

S.  77°  30'  E. 

40.7 

“ breast  of  tunnel. 

Breast. 

Centre  of 

In  shaft. 

-|-  10®  20' 

S.  65°  00'  E. 

73 

In  shaft. 

“ centre  of  shaft. 

Shaft. 

- 75°  30' 

N.  2°  15'  W. 

102.12 

“ point  in  shaft  at  tunnel 
level. 

The  depth  of  the  tunnel  at  the  breast  is  determined  as  in  the  first  example. 
The  vertical  distance  between  the  point  over  the  breast  and  the  mouth  of  the 
shaft  is  determined  by  the  equation; 

Difference  of  elevation  = 73  X tan  10*  20'  = 13.31  ft. 

Add  this  to  the  depth  of  the  tunnel  at  the  breast  and  we  have  13.31  -|-  85.56  = 
98.87  ft.,  the  vertical  depth  at  which  the  tunnel  will  cut  the  shaft.  With  this 


MINING  SURVEYING. 


351 


depth  and  the  pitch  of  the  shaft,  14°  30',  we  obtain  the  depth  (distance  along  the 
shaft)  at  which  the  tunnel  will  cut  the  shaft,  measured  along  the  dip,  and  also 
the  horizontal  distance  from  the  instrument  station,  which  is  the  intersection 
of  the  centre  line  of  the  shaft  with  the  surface,  to  the  point  of  intersection,  by 
the  following  equations: 


Depth  measured  on  the  dip  = 98.87  X sec  14°  30'  = 102.12  ft. 

Horizontal  distance  = 98.87  X tan  14®  30'  = 25.57  ft- 

Set  a stake  N.  2”  15'  W.  25.57  ft.  from  the  instrument,  and  connect  it  with  the 
point  over  the  breast  of  the  tunnel,  and  we  have  the  course  and  distance 
from  the  breast  of  the  tunnel  to  the  line  of  survey  down  the  shaft,  S.  85°  21' 
E.  65.22  ft. 

273.  To  survey  a Mine,  with  its  Shafts  and  Drifts. — 

Set  up  the  instrument  at  the  top  of  the  main  shaft,  and  after 
having  first  obtained  the  meridian,  take  the  bearing-distance 
and  vertical  angle  to  the  point  selected  for  the  first  station  in 
the  shaft.  The  distance  is  to  be  measured  on  a direct  line  be- 
tween the  stations,  and  its  horizontal  and  vertical  components 
afterwards  calculated  from  the  data  obtained.  The  stations 
in  the  shaft  are  to  be  selected  with  a view  to  the  extension  of 
the  survey  into  the  different  levels  and  down  the  shafts,  and, 
as  in  case  of  other  underground  surveys,  it  is  well  to  explore 
the  mine  ahead  of  the  work,  that  the  stations  may  be  selected 
advantageously. 


The  field-notes  of  the  survey  of  a mine  are  here  given  for  illustration.  The 
horizontal  and  vertical  components  of  the  distances  measured  down  the  shafts 
can  be  obtained  by  the  use  of  a table  of  natural  sines  and  cosines.  (Figs.  100 
and  loi.) 


352 


SURVEYING. 


FIELD.NOTES. 


Sta. 

Course. 

Dist. 

Vertical 

Angles. 

N.  53°  57'  W. 

54-1 

t 

0 

a 

N.  34«  52'  E. 

57-6 

0®  00' 

S.  25»  13'  W. 

60.8 

0°  00' 

N.  57»  46'  W. 

60.0 

- 66®  24' 

3 

N.  30®  30'  E. 

73-0 

0®  00' 

N.  69®  23'  W. 

47-5 

- 77®  30' 

4 

N.  28®  24'  E. 

75 

0®  00' 

S.  19°  59'  W. 

90 

0®  00' 

N.  71®  15'  W. 

55 

— 78®  00' 

5 

N.  37*  20'  E. 

75 

0°  00' 

S.  28®  43'  W. 

64 

0®  00' 

N.  72®  oi'  W. 

55-6 

1 

0 

0 

to 

0^ 

6 

S.  71®  50'  E. 

5-1 

0°  00' 

S.  71®  50'  E. 

8.1 

0 00' 

7 

N.  66°  15'  W. 

46.8 

— 85®  00' 

9 

N.  32®  50'  E. 

60 

0®  00' 

S.  24®  00'  W. 

51 

0°  00' 

N.  55“  03'  W. 

40 

- 86®  46' 

10 

N.  34®  15'  E. 

39 

0°  00! 

S.  37“  45'  W. 

55 

0®  00' 

N.  88®  30'  W. 

48.6 

— 81®  00' 

11 

N.  34®  00'  E. 

54 

0®  00' 

S.  24®  00'  W. 

^5 

0®  00' 

Hori- 

zontal 

Com- 

ponent. 


^3-37 


10.76 


”•43 


2.25 


7.60 


Vertical 

Com- 

ponent. 


54-98 


46-37 


S3 -So 


54.67 


46.62 


39-93 


48.0 


Remarks. 


Begin  at  Sta.  i at  top  of  shaft. 

To  Sta.  2 at  1st  level  in  shaft. 

" air-shaft  at  end  of  ist  level. 

“ centre  of  bottom  of  discov- 
ery-shaft, 50  ft.  deep. 

“ Sta.  3 at  2d  level  in  shaft. 

‘‘  breast  of  2d  level,  running 
N.E.  Second  level  run- 
ning S.W.  filled  with  de- 
bris, not  accessible. 

“ Sta.  4 at  3d  level  in  shaft. 

“ breast  of  3d  level,  running 
N.E. 

“ breast  of  3d  level,  running 
S.W. 

“ Sta.  5 at  4th  level  in  shaft. 

“ 4*^^  level,  running 

“ breast  of  4th  level,  running 
S.W. 

“ Sta.  6 at  5th  level  in  shaft. 
The  vein  here  divides: 
the  shaft  follows  the  por- 
tion to  the  south.  The 
shaft  is  chambered  out  at 
this  point,  being  10  ft. 
wide  S.W. , and  20  ft.  long 
S.E.  from  Sta.  6. 

“ Sta.  7 at  top  of  shaft  in 
chamber. 

“ Sta.  8 opposite  drift,  run- 
ning S.  24*  15'  W.,  102  ft. 

“ Sta.  9 at  6th  level  in  shaft. 

“ breast  of  6th  level,  running 
N.E. 

“ breast  of  6th  level,  running 
S.W. 

“ Sta.  10  at  7th  level  in  shaft. 

“ breast  of  7th  level,  running 
N.E. 

“ breast  of  7th  level,  running 
S.W. 

“ Sta.  II  at  8th  level,  at  bot- 
tom of  shaft. 

“ breast  of  8th  level,  running 
N.E. 

“ breast  of  8th  level,  running 
S.W. 


MINING  SURVEYING. 


353 


Note. — The  width  of  the  levels  of  this  mine  are  about  four  feet.  The 
dimensions  of  the  shaft  are  4'  X 10'.  The  line  of  survey  down  the  shaft  was 


Fig.  ioo. 


run  about  2 ft.  from  the  north  end  of  the  shaft.  In  the  levels  the  line  of  survey 
was  to  the  centre  of  the  breast. 


23 


354 


SURVEYING. 


Fig.  loi  shows  the  plan,  and  Fig.  too  the  longitudinal  and  transverse  sec- 
tions of  the  mine  as  plotted  from  the  field-notes.  The  plan  is  plotted  from  the 
courses  and  horizontal  components  of  the  measurements  in  the  shaft  and  levels, 
as  projected  upon  a horizontal  plane. 

The  longitudinal  section  is  platted  from  the  courses  and  vertical  components 
of  the  measurements  in  the  shaft,  and  the  horizontal  measurements  in  the 
levels  as  projected  upon  a vertical  plane  passing  through  Station  i and  at  right 


PLAN 


* Fig.  ioi. 

angles  to  a vertical  plane  passing  through  Stations  i and  ii,  upon  which  the 
transverse  section  is  plotted  as  projected  from  the  vertical  angles,  courses,  and 
measurements  in  the  shaft. 

Thus  it  will  be  seen  that  for  the  full  representation  of  an  underground  sur- 
vey, showing  the  relative  position  of  the  parts  to  each  other,  three  planes  are 
necessary, — two  vertical  planes  at  right  angles  to  each  other,  and  a horizontal 
plane. 


274.  Conclusion. — The  above  examples  comprehend  some 
of  the  more  general  cases  arising  in  the  practice  of  mining 


MINING  SURVEYING. 


355 


surveying;  any  other  cases  which  may  arise  will  be  found  to 
be  modifications  or  combinations  of  these.  The  problem  to  be 
considered  can  be  solved  by  an  application  of  the  principles 
therein  embraced,  which  the  ^surveyor  will  find  useful,  also,  in 
solving  problems  of  mining  engineering  relating  to  the  meas- 
urement of  ore  reserves,  development,  and  systems  of  working. 
It  has  been  shown  that  the  following  out  of  the  underground 
workings  of  a mine  corresponds  to  traversing  when  elevations 
are  carried  by  means  of  vertical  angles,  as  was  fully  described 
in  the  chapter  on  topographical  surveying.  The  notes  are 
also  reduced  in  the  same  manner. 

It  has  been  the  object  of  this  chapter  to  present  the  subject 
of  mining  surveying  in  as  simple  a form  as  possible,  and  divest 
it  of  all  features  which,  although  they  may  give  it  a distinctive 
aspect,  serve  only  to  render  it  more  complex  and  give  the 
reader  an  idea  of  difficulties  which  are  only  imaginary. 

It  is  useless,  also,  for  the  mining  surveyor  to  encumber  him- 
self with  many  paraphernalia.  Good  work  can  be  done  with 
a mining  transit  provided  with  an  extra  telescope  for  vertical 
pointings,  one  or  two  short  rods,  and  a reliable  steel  tape,  all 
of  which  can  be  carried  by  the  surveyor  on  horseback  over  the 
rough  mountainous  roads.  Any  other  adjuncts  can  be  im- 
provised or  be  found  at  any  well-conducted  mine,  and  would 
prove  more  burdensome  than  useful. 


CHAPTER  XII. 


CITY  SURVEYING. 

275.  Land-surveying  Methods  inadequate  in  City  Work. 

— The  methods  described  in  the  chapter  on  Land-surveying 
are  inadequate  to  the  needs  of  the  city  surveyor.  The  value 
of  the  land  involved  in  errors  of  work,  with  such  a limit  of  er- 
ror as  was  there  found  practicable  (see  art.  175),  is  so  great  as 
to  justify  an  effort  to  reduce  this  limit.  Comparing  the  value 
of  a given  area  of  the  most  valuable  land  in  large  cities  with 
the  value  of  a like  area  of  the  least  valuable  land  which  a sur- 
veyor is  ever  called  upon  to  measure,  the  ratio  is  more  than  a 
million  to  one. 

This  view  is  emphasized  by  the  manner  of  use.  On  farm 
lands  the  most  valuable  improvements  are  placed  far  within 
the  boundary-lines,  but  the  owner  of  the  city  lot  is  compelled 
by  his  straitened  conditions  to  place  the  most  costly  part  of 
his  improvements  on  the  limit-line.  His  neighbor’s  wall  abuts 
against  his  own.  The  surveyor,  who  should  retrace  this  line 
and  make  but  a small  difference  of  location,  would  get  his 
clients  and  himself  into  trouble.  Both  the  value  of  the  land 
and  the  manner  of  its  use  demand  increased  care.  The  modi- 
fications of  the  methods  used  in  land-surveying  to  meet  the 
requirements  of  work  in  the  city  will  be  treated  in  this  chapter. 
Much  of  the  work  described  furnishes  data  for  the  solution  of 
engineering  problems,  but  the  obtaining  of  the  facts  falls  en- 
tirely within  the  definition  of  surveyor’s  work. 


CITY  SURVEYING, 


357 


276.  The  Transit  is  used  exclusively,  but  the  common  pat- 
tern may  be  very  materially  modifi^ed  with  obvious  advantage. 
Seeing  that  the  magnetic  needle*  is  never  precise  and  seldom 
correct,  it  should  be  wholly  discarded  in  the  construction  of 
the  city  surveyor’s  transit.  The  verniers  can  then  be  placed 
under  the  eye,  the  bubbles  can  be  removed  from  the  standards 
and  placed  upon  the  plate  of  the  alidade,  and  the  standards 
themselves  can  be  more  firmly  braced.  By  these  changes  a 
steadier  and  more  convenient  instrument  is  secured,  when  the 
useless  and  somewhat  costly  appendage  of  a needle-box  is  out 
of  the  way.  The  adjustable  tripod  head  and  the  levelling 
attachment  are  always  convenient.  For  topographical  work, 
the  vertical  circle,  or  a sector,  and  stadia  wires  are  essential, 
otherwise  the  methods  used  must  be  primitive.  The  ther- 
mometer which  is  needed  in  order  to  make  the  proper  correc- 
tions for  temperature  may  be  conveniently  attached  to  one  of  the 
standards  facing  the  eye-piece  of  the  telescope.  The  danger 
of  breaking  the  tube  while  handling  the  instrument  may ‘be 
obviated  by  a guard  sufficiently  deep  to  protect  the  bulb,  made 
open  on  the  side  toward  the  observer. 

277.  The  Steel  Tape  is  generally  used  for  measuring.  The 
legal  maxim  that  “ distances  govern  courses,”  when  interpreted, 
means  that,  using  customary  methods,  the  intersection  of  two 
arcs  of  circles,  centres  and  radii  being  known,  is  a more  definite  lo- 
cation of  a point  than  the  intersection  of  two  straight  lines  whose 
origin  and  direction  are  likewise  known.  The  fact  is,  the  inter- 
sections are  not  more  definite.  The  maxim  grew  into  authority 
when  the  compass  was  pitted  against  the  chain.  With  the 
transit  to  define  directions  of  courses,  and  the  chain  still  to 
measure  the  distances,  such  a maxim  would  not  have  voiced 
the  results  of  experience,  but  would  have  been  sheer  nonsense. 

* The  needle  finds  its  proper  place  where  checks  are  not  so  abundant,  and  in 
classes  of  work  in  which  a close  and  rapid  approximation  is  of  more  value  than 
precision. 


358 


SUR  VE  YING. 


The  ordinary  chain  has  too  many  gaping  links,  and  the  brazed 
chain  too  many  wearing  surfaces,  to  be  kept  in  very  close  ad- 
justment to  standard  length.  Its  weight  is  such  as  to  make  the 
“normal  tension”  (see  p.  375)  impracticable;  hence  the  effect 
of  slight  variations  of  pull  is  much  more  marked  than  if  the 
tape  is  used.  Graduated  wooden  rods  were  used  until  i860  to 
1870.  They  were  unwieldy  when  twenty  feet  long,  and  were 
still  so  short  that  the  uncompensated  part  of  their  compen- 
sating errors  was  a matter  of  considerable  moment.  Every 
time  the  pin  is  stuck  or  a mark  made  at  the  forward  end  of  the 
tape  or  rod,  the  work  is  a matter  of  skill  and  involves  an  error 
dependent  on  the  degree  of  skill  attained.  When  the  measure 
is  brought  forward,  its  proper  adjustment  in  the  new  position 
is  a matter  requiring  skill.  These  errors  are  compensating,  but 
the  resultant  is  not  zero.  The  use  of  the  plumb-line  is  another 
source  of  compensating  errors  which  are  reduced  by  an  increase 
of  length  in  the  measure.  First,  the  number  of  applications 
varies  inversely  as  the  length  of  the  measure  ; second,  using  the 
rod,  it  was  necessary  to  work  to  the  bottom  of  ravines  and  gul- 
lies and  then  work  up  again  ; now  the  long  tape  spans  them  at 
a single  application.  The  minus  errors  due  to  imperfect  align- 
ment and  inaccurate  levelling  of  the  two  ends  have  a greater 
percentage  of  effect  when  the  measure  is  short  than  when  it  is 
long.  The  longer  tape  brings  with  it  some  other  sources  of 
error.  When  used  suspended  at  the  ends  there  is  a minus 
error  on  account  of  the  sag  of  the  intermediate  parts,  and  a 
plus  error  from  elongation  due  to  tension  ; there  is  also  expan- 
sion by  heat,  which  produces  an  error  which  may  be  plus  or 
minus  as  the  temperature  at  the  time  and  place  is  above  or 
below  that  for  which  the  tape  is  tested.  The  effect  of  sag 
increases  very  nearly  as  the  cube  of  the  length  when  the  ten- 
sion is  constant.  When,  to  counteract  this  increase,  the 
pull  is  made  greater  than  a man  can  apply  uniformly  under 
all  conditions — at  his  feet  or  above  his  head — there  come 


CITY  SURVEYING. 


359 


irregularities  from  this  cause.  The  limit  of  length  of  tape 
which  it  is  practicable  to  use  will  be  determined  by  the  condi- 
tions of  the  work  and  should  be  such  that  the  increase  of 
length  involves  greater  error  than  it  eliminates.  On  account 
of  convenience  in  keeping  tally,  50-foot  and  lOO-foot  lengths 
are  generally  used.  In  a level  country  the  lOO-foot  tape  is  pre- 
ferred. 

There  are  tapes  made  with  the  purpose  to  eliminate  the 
errors  which  arise  from  the  free-hand  pull,  the  inclination  of 
the  tape,  and  the  temperature.  As  seen  by  the  writer,  they 
carry  a spring-balance  marked  for  a pull  of  ten  pounds,  a 
bubble  adjusted  to  the  inclination  of  the  end  of  the  tape  at 
that  pull,  and  a thermometer  graduated  to  such  a scale  that 
each  division  corresponds  to  one  turn  of  a screw  adjusting 
the  whole  length  of  the  tape  to  the  changes  of  temperature. 
The  whole  was  connected  by  rings  and  swivels,  eight  or  nine 
wearing  surfaces,  some  of  them  conical,  to  a tape  which  carried 
no  graduation.  The  effort  is  laudable ; but,  probably  on  ac- 
count of  the  number  and  form  of  the  wearing  surfaces,  they 
have  not  yet  met  with  general  favor.  Further  progress  may 
be  made  in  this  direction. 

LAYING  OUT  A TOWN  SITE. 

278.  Provision  for  Growth. — Cities  grow.  It  is  very  rare 
that  the  considerations  which  should  have  governed  have  been 
given  any  place  in  determining  upon  the  plan  of  the  original 
town.  The  considerations  first  in  importance  are  topographi- 
cal. What  are  the  natural  lines  along  which  business  will  tend 
to  distribute  itself?  To  what  form  of  subdivision  can  it  adapt 
itself  with  the  least  resistance  ? Where  is  the  best  harbor, 
the  lake  or  river  front,  or  the  railway  line  ? Ordinarily  the 
land  immediately  adjoining  such  natural  features  is  not  best 
used  when  used  as  a street,  but  when  occupied  by  private 


36o 


SURVEYING. 


docks,  or  along  a railway  by  warehouses  and  factories  having 
switching  facilities  without  crossing  public  streets.  The 
streets  parallel  to  such  lines  should  be  of  ample  width, 
easy  grade,  and  continuous  but  not  necessarily  straight  align- 
ment. Much  of  the  heavy  hauling  will  be  along  such  streets. 
In  the  business  part  of  the  town  the  cross-streets  should 
be  so  frequent  as  to  make  the  blocks  approximately  square. 
In  the  residence  portion  alternate  streets  in  one  direction  may 
with  advantage  be  omitted:  this  saves  the  expense  of  unneces- 
sary streets,  and  permanently  lightens  the  burden  of  taxation. 
Which  fronts  are  on  all  accounts  most  desirable  in  the  par- 
ticular locality  will  determine  in  which  direction  the  blocks 
should  be  longest. 

279.  Contour  Maps. — Another  phase  of  topography  de- 
mands attention.  The  sites  of  suburban  towns  may  generally 
be  best  handled  by  laying  out  streets  and  lot  lines  in  conformity 
to  the  undulations  of  the  ground.  Additions  to  the  city  may 
also  have  characteristic  features  that  can  be  preserved  with 
advantage.  For  all  such  cases  a contour  map  is  very  useful 
to  one  who  is  able  to  interpret  it.  The  making  of  all  the 
ground  available,  and  sightly  points  accessible,  and  at  the  same 
time  so  locating  the  streets  as  to  secure  economical  grades, — in 
short,  the  judicious  handling  of  the  whole  subject  is  facili- 
tated by  the  study  of  the  contour  map. 

280.  The  Use  of  Angular  Measurements  in  Subdivi- 
sions.— Shall  subdivision  lines  be  located  by  an  angle  with  the 

street  on  which  the  lots  front  or  by 
distances  from  the  next  cross-street  ? 
Must  distances  govern  courses,  what- 
ever methods  are  used  ? Let  us  sup- 
pose, for  illustration,  that  it  is  re- 
^ quired  to  locate  lot  o in  the  accom- 

FiG.  102.  . 1 T^.  \ O 

panying  sketch  (Fig.  102).  Suppose, 
farther,  that  it  is  possible  to  measure  each  of  the  lines  ab 


eiA- 


-400- 


' 50 

2 

3 

4 

5 

6 

7 

8 

9 S 
/ 

-A 

-A 

s.SO'i 

” 

” 

.» 

II 

»» 

II 

CITY  SURVEYING. 


361 


and  dc  with  a maximum  error  of  i in  5000  and  that  the 
conditions  are  such  as  to  produce  opposite  errors  in  the 
two  lines.  Then,  ist,  the  resulting  error  in  locating  the  line 
be,  i.e.  {ab  — dc')  will  be  3-0V0  X 400  X 2 = 0.16  feet.  The 
sine  of  the  angle  by  which  the  angle  A'  differs  from  A will  be 
= .00107.  Hence  the  change  of  direction  on  account  of 
the  errors  in  measurement  is  3f  minutes.  2d,  the  line  ef 
must  be  distant  from  ab^\Y.  150  feet  ^'550  feet,  in  order 
that,  under  like  conditions,  if  it  is  measured  instead  of  dc,  the 
change  in  direction  shall  not  exceed  one  minute.  Or  the  loca- 
tion may  be  made  by  measuring  the  line  ab,  or  a line  near  to 
it  where  favorable  conditions  exist,  and  then  repeating  ba 
the  same  man  being  fore-chainman ; the  principle  of  reversal 
is  thus  applied  to  this  measurement.  Then  measuring  A'  = A 
and  repeating  the  angle,  reading  both  verniers,  the  error  is 
brought  within  the  maximum  error  in  the  pointing  power  of 
the  instrument.  In  order  to  locate  be  from  ab  parallel  to  ad, 
two  monuments  marking  the  line  ab  need  to  be  known.  The 
other  method  requires  also  a monument  locating  the  line  ae. 
It  thus  appears  that  when  the  side-lines  of  lots  are  located 
perpendicular,  or  at  any  other  known  angle  with  the  street 
upon  which  the  lot  fronts,  it  is  susceptible  of  more  accurate 
location  than  by  two  (front  and  rear)  measurements,  unless  the 
usual  limit  of  error  can  be  greatly  reduced.  While  it  is  not 
likely  that  maximum  errors  of  opposite  character  will  fall  to- 
gether affecting  the  work  on  the  same  lot,  it  is  quite  as  im- 
probable that  the  maximum  error  in  measuring  an  angle 
should  vitiate  the  work  of  the  transit.  It  is  probably  quite  as 
easy  to  reduce  the  maximum  error  in  measuring  an  angle  to 
half  a minute  as  it  is  to  keep  the  maximum  error  in  measur- 
ing distances  down  to  i in  10,000. 

281.  Laying  out  the  Ground. — The  work  of  putting  the 
plan  upon  the  ground  is  a very  important  one.  This  is  about 
the  worst  possible  place  to  do  hurried  and  inaccurate  work. 


362 


SURVEYING. 


Fences  or  other  styles  of  marking  possession  which  limit  the 
contour  map  cannot  be  relied  upon  as  defining  the  property- 
lines.  These  lines  must  be  accurately  located  in  relation  to 
the  streets  of  the  town  or  of  the  addition,  in  order  to  make 
practicable  such  exchanges  or  sales  as  may  be  necessary  to  ad- 
just property-lines  to  the  new  block-lines.  This  method  is 
preferable  to  that  which  adjusts  block-lines  to  the  original 
property-lines.* 

As  a framework  for  the  whole  survey  an  outline  figure, 
generally  a quadrilateral,  of  sufficient  dimensions,  and  so 
placed  that  it  can  be  permanently  marked  with  monuments 
which  will  remain  accessible  when  the  town  is  built  up,  should 
be  located  with  especial  care.  All  lines  should  be  measured, 
all  angles  observed,  and  all  practicable  checks  introduced. 
This  figure  must  close  absolutely  ; that  is,  the  record  of  the 
work  when  completed  should  be  mathematically  consistent. 
Unreasonable  errors  are  to  be  eliminated  by  retracing  the  work. 
In  the  adjustment  which  distributes  the  remaining  errors  each 
part  of  the  work  should  be  weighted  (art.  174,  Rule  2),  for  it 
is  very  rare  that  a land-survey  is  completed  under  such  con- 
ditions that  the  man  who  does  the  work  would  be  justified, 
while  these  conditions  are  fresh  in  his  mind,  in  assuming  that 
the  probability  of  error  is  alike  at  all  points.  The  angles  ad- 
mit of  adjustment  independently  of  the  length  of  the  lines. 
That  distribution  of  the  angular  errors  which  reduces  the  errors 
of  measurement  to  a minimum  has  such  weight  that  it  can 
be  overruled  only  by  the  most  positive  evidence  that  the  cor- 


* In  some  places  this  idea  of  the  private  interest  of  the  proprietor,  some- 
times private  spite,  is  carried  to  such  an  extent  that  it  would  seem  that  each 
man’s  farm  or  garden  patch  was  especially  fitted  to  be  a town  by  itself,  laid 
out  with  utter  disregard  to  the  towns  which  others  are  in  like  manner  laying 
out  upon  adjacent  farms.  In  this  practice  the  interests  of  the  public  for  all 
time  are  neglected  in  order  to  secure  a doubtful  advantage  for  one.  Where  the 
custom  prevails  it  is  better  honored  in  the  breach  than  in  the  observance. 


CITY  SURVEYING. 


363 


rections  so  indicated  cannot  be  the  true  ones.  The  distances 
are  then  adjusted  to  the  angles  so  determined.  The  re- 
mainder of  the  work  of  the  subdivision  is  checked  upon  the 
adjusted  outline,  reasonable  errors  being  distributed  and  un- 
reasonable ones  retraced. 

282.  The  Plat  to  be  geometrically  consistent.— The 

necessity  that  the  recorded  plat  should  be  consistent  lies  in 
the  use  that  is  to  be  made  of  it.  A parcel  of  ground  de- 
scribed by  reference  to  the  plat  of  record  should  have  but  one 
location,  not  any  one  of  two  or  more  possible  locations,  as  is 
the  case  when  the  plat  contains  errors  on  its  face.  In  the 
course  of  years  the  lines  of  such  parcels  will  be  retraced  proba- 
bly many  times,  at  one  time  by  one  method,  at  another  time 
by  another  equally  in  accord  with  the  plat.  If  the  plat  is  not  > 
consistent  with  itself  and  with  the  monuments  upon  the 
ground,  this  error  will  be  pretty  sure  to  find  its  way  into  the 
lot  location.  When  the  fault  is  with  the  plat,  it  matters  not 
how  the  monuments  are  placed  upon  the  ground  ; they  cannot 
mark  the  chief  points  and  all  agree  in  such  a way  that  if  any 
two  remain  and  the  others  are  lost  the  relocation  will  in  every 
case  be  the  same.  But  this  is  just  what  the  plat  is  for — to 
make  a public  record  of  the  relation  of  each  part  of  the  sub- 
division to  every  other. 

283.  Monuments. — How  many  monuments  shall  be  lo- 
cated, and  where  shall  they  be  placed  ? What  material  shall 
be  used  and  how  set?  Answering  the  first  question,  it  is  plain 
that  no  more  work  should  be  attempted  than  can  be  done  well. 
Better  one  point  and  an  azimuth  than  points  everywhere  and 
no  two  agreeing  either  in  distance  or  direction  with  the  rela- 
tion described  by  the  plat.  But  so  much  should  be  done  well 
that  the  labor  of  locating  any  point  in  the  subdivision  from 
existing  monuments  shall  not  be  excessive.  The  points 
chosen  for  placing  monuments  should  be  such  as  will  continue 
to  be  accessible  and  will  not  be  ambiguous.  The  centre  lines 


364 


SURVEYING. 


of  intersecting  streets  are  sometimes  marked,  giving  one  monu^ 
ment  to  each  intersection  ; others  choose  the  side-lines,  giving 
four  monuments  to  each  intersection  of  streets.  If  the  blocks 
are  so  long  that  intermediate  points  are  desirable,  points  on 
the  ridges  should  be  selected. 

Stone  is  more  often  chosen  than  any  other  material ; iron 
bars,  gun-barrels,  gas-pipe,  etc.,  are  sometimes  used,  driven 
with  a sledge  ; cedar  posts,  say  4"  X 4^,  are  quite  durable,  and 
hard-burned  pottery  is  sometimes  used.  Whatever  material 
is  chosen,  the  foundation,  which  should  be  flat — not  pointed — 
must  reach  below  frost;  and  the  centre  of  gravity  is  kept  as  low 
as  possible,  so  that  there  shall  be  no  tendency  to  settle  out  of 
place  when  the  ground  is  soft  in  the  spring.  When  the  tops 
are  much  above  the  surface  of  the  ground,  there  is  a liability 
that  they  may  be  displaced  by  traffic.  Probably  the  surveyor 
does  not  see  any  traffic,  or  the  prospect  of  it,  when  he  is  doing 
his  work,  but  the  traffic  must  come  before  the  work  of  the 
monument  can  be  spared.  It  is  better  to  bury  the  stone  wholly 
and  indicate  where  to  dig  for  it  by  bearings  than  to  run  the 
risk  of  losing  the  whole  work  through  indiscretion  in  placing 
the  monument  that  marks  it.  In  situations  where  every  rain 
storm  produces  a slight  fill  it  is  safe  to  place  the  top  consider- 
ably higher  than  would  otherwise  be  reasonable.  The  stones 
to  be  set  are  so  placed  in  the  excavation,  with  the  heavy  end 
down,  that  when  the  top  is  in  the  proper  position  and  before  any 
earth  is  refilled  there  is  no  tendency  to  fall  in  any  direction  ; then 
while  the  earth  is  being  refilled  and  thoroughly  tamped  about 
the  stone,  the  top  is  kept  in  place.  It  is  better  that  the  mark 
denoting  the  point  for  which  the  stone  stands  should  be  cut 
upon  before  it  is  placed  in  the  ground.  When  this  is  done,  if 
the  mark  is  worn  off  by  traffic  or  knocked  off  by  accident,  the 
centre  of  that  portion  of  the  stone  which  remains  is  a very 
close  approximation  to  the  original  point.  A slovenly  way  of 
slighting  this  work  is  to  tumble  the  stone  into  the  excavation. 


CITY  SURVEYING. 


365 


fill  around  it  pretty  much  as  it  happens,  push  it  to  one  side  or 
another  so  that  the  point  will  come  somewhere  on  the  top,  and 
then  cut  the  mark  wherever  the  point  comes.  Stones  set  in  this 
way  are  liable  to  settle  out  of  place  after  the  first  heavy  rain, 
while  frost  and  rain  keep  up  their  work  till  the  stone  lies  flat 
upon  its  side.  If  by  chance  it  should  keep  its  place  pretty 
well  and  the  mark  becomes  defaced,  it  might  as  well  be  any 
loose  bit  of  rock  as  a set  stone,  for  its  centre  gives  no  idea  of 
where  the  mark  was  placed.  No  one  should  be  trusted  to  set 
corner-stones  unwatched  who  is  not  familiar  with  the  work 
and  thoroughly  reliable. 

Points  are  preserved  temporarily  by  wooden  stakes  driven 
flush  with  the  ground.  The  point,  preserved  by  offsets  while 
the  stake  is  being  driven,  is  marked  by  a nail.  Witness-stakes 
driven  alongside  and  standing  above  grass  and  weeds  assist  in 
finding  the  stakes  when  wanted.  Made  of  half-decayed  soft 
wood,  e.g.,  old  fence-boards,  such  stakes  will  hardly  last  a 
season ; while  durable  wood,  well  seasoned,  will  last  much 
longer  than  any  driven  stake  can  be  relied  upon,  since  it  does 
not  go  below  frost,  and  is  liable  to  be  pushed  by  a passing 
wheel  or  be  otherwise  disturbed  when  the  ground  is  soft. 

284.  Surveys  for  Subdivision. — The  purpose  of  making  a 
survey  before  recording  a plat  of  a subdivision  is  twofold, — 
first,  to  get  the  information  which  it  is  desirable  to  record ; 
second,  to  leave  such  monuments  as  will  make  it  easy  to  locate 
any  portion  when  desired.  The  recorded  plat  should  show 
sufficient  facts  to  determine  the  relations  of  every  part  to  the 
whole,  and  these  relations  should  be  shown  by  methods  which 
involve  the  minimum  of  error,  i.e.,  giving  a location  which  may 
be  retraced  with  least  possible  doubt.  The  current  practice 
falls  short  of  this  standard  at  some  points  which  are  worthy  of 
note. 

{a)  Surveyors  seem  to  have  no  doubt  of  the  ability  of  their 
field-hands  to  measure  a line,  but  very  seriously  doubt  their 


366 


SURVEYING. 


own  ability  to  measure  an  angle.  Angles  are  measured  dur- 
ing the  progress  of  the  work  and  are  used  for  determining  the 
lengths  of  lines  ; these  lengths  are  then  made  a part  of  the 
record,  while  the  angles  which  determined  them  are  omitted. 
Apparently  some  things  which  arc  dependent  have  become 
more  certain  and  fixed  than  that  upon  which  they  depend.  A 
proper  record  of  angles  would  show  what  lines  are  straight  and 
where  defictions  are  made.  Defleections  which  are  sufficient 
to  very  seriously  affect  the  position  of  a brick  wall  do  not  show 
on  the  scale  of  the  recorded  plat.  For  example,  an  addition  to 
a town  extends  from  Fifth  Street  to  Twelfth  Street  ; extreme 
points  are  well  established,  but  intermediate  monuments  are 
missing;  and  it  is  required  to  establish  at  Eighth  Street  the 
line  of  a street  which  a ruler  applied  to  the  recorded  plat  sug- 
gests is  a straight  line.  Custom  approves  that  in  such  a case 
the  surveyor  should  try  a straight  line,  there  being  a mild  pre- 
sumption in  its  favor;  but  if  his  straight  line  agrees  with  one 
wall  and  disagrees  with  two  walls  and  a fence,  he  had  better 
look  further  before  he  comes  to  a decision.  No  such  doubt 
could  have  existed  if  the  recorded  plat  had  been  properly  made. 

(b)  Very  few  recorded  plats  show  what  stones  have  been 
set  by  the  surveyor,  or  indeed  indicate  that  he  has  any  knowl- 
edge that  such  monuments  may  ever  be  useful.  If  the  custom 
were  once  established  of  noting  upon  the  record  the  location 
and  description  of  monuments,  any  monument  found  during  a 
resurvey,  but  not  shown  on  the  record,  would  be  discredited. 
As  matters  now  stand  it  must  be  proved  incorrect  to  be  dis- 
credited— a thing  not  always  easy,  for  a system  of  quadrilat- 
eral blocks  whose  angles  are  not  recorded  and  whose  street 
lines  are  not  necessarily  straight  is  not  theoretically  very  rigid. 

(c)  Many  plats  require  measurements  to  be  made  along 
lines  which  are  easily  measured  while  the  land  is  vacant,  but 
which  will  become  inaccessible  as  soon  as  the  property  is  built 
up.  The  obstacles  to  be  overcome  before  the  result  can  be 


CITY  SURVEYING. 


367 


reached  by  the  method  described  on  the  record  will  each  add 
to  the  doubt  of  the  accuracy  of  that  result.  There  are  many 
ways  in  which  plats  are  made,  which  are  all  justly  subject  to 
this  criticism.  Two  examples  will  suffice.  Irregularly  shaped 
blocks  are  sometimes  treated  as  in  the  annexed  sketch,  Fig. 
103.  The  outline  is  subdivided  mechanically,  and  proportional 


distances  are  given  on  interior  lines  which  are  not  consistent 
with  any  trigonometrical  relation  of  the  exterior  lines,  much 
less  with  that  which  does  exist  but  is  not  recorded.  The  point 
X has  nine  distinct  locations  directly  from  the  plat.  On  the 
theory  that  ah  and  cd  are  straight  lines,  their  intersection  gives 
one;  ab  straight,  the  distances  ax  and  bx  give  each  one;  cd 
straight,  the  distances  cx  and  dx  give  two.  Combine  the  dis- 
tances ax  and  cx,  bx  and  cx,  etc.,  and  get  four  more.  But  this 
is  not  all,  for  the  point  x stands  related  to  each  of  the  ten  other 
points  along  the  line  ab,  and  each  of  these  has  also  nine  loca^ 
tions  which  accord  with  the  plat,  and  our  point  x may  be  lo^ 
cated  from  either  of  them  or  any  combination  of  them  when 
they  have  been  located  by  any  of  the  methods  described. 

Besides  the  difficulty  of  determining  how  interior  points 
should  be  located,  this  fan-like  subdivision  wastes  ground  in 
each  lot  which  results  in  wedge-shaped  remnants  about  the  build- 
ings, which  remnants  would  be  valuable  if  thrown  together  into 
the  corners,  thus  keeping  the  remaining  lots  rectangular  at  the 


368 


SUR  VE  YING. 


front.  The  attempt  to  reach  a rectangular  front  sometimes 
fails  through  inattention  to  very  simple  matters,  as  in  Fig.  104. 
Here  no  angles  are  recorded.  The  rear  corners  of  the  lots  are 
located  along  the  line  ab  by  distances  from  aor  b\  but  the 
record-depths  do  not  fall  upon  a straight  line.  The  line  ab 
should  bisect  the  angle  between  the  block-lines  or  be  parallel 
to  such  bisection  in  order  that  with  a constant  distance  along 
ab  common  to  the  series  of  lots  on  each  side  of  that  line  their 


angles  with  their  respective  fronts  may  remain  constant.  In 
the  case  given  every  lot-line  has  an  angle  with  the  block-line 
upon  which  it  fronts  different  from  that  of  every  other  lot-line, 
and  all  dependent  on  some  block-angle  which  is  not  recorded. 
If  it  is  not  desirable  to  bisect  the  block  by  the  line  ab,  its  di- 
rection may  be  chosen  as  desired,  the  distances  along  it  are 
fixed  by  the  fronts  on  one  and  the  angular  divergence  from 
that  side  which  is  chosen,  and  the  lot  fronts  on  the  other  side 
of  the  block  must  be  correspondingly  increased  or  diminished. 

When  alleys  are  laid  out  in  a block  so  that  the  interior  lines 
are  accessible,  it  is  very  rare  that  after  the  block  is  improved 
these  lines  can  be  measured  under  the  same  conditions  as  the 
fronts.  If  alleys  are  not  laid  out,  the  difficulties  are  usually 
much  greater.  Location  of  lot-lines  by  angle  from  the  front  is 


CITY  SURVEYING. 


369 


undoubtedly  the  most  uniform  and  workmanlike  method  avail- 
able to  the  surveyor.  Hence,  distances  on  the  rear  lines  of  the 
corner  lots  should  be  omitted  from  the  record,  if  their  presence 
would  leave  any  doubt  as  to  which  method  of  location  is  in- 
tended. It  is  not  customary,  nor  is  it  desirable,  that  lot-lines  or 
distances  should  be  determined  upon  the  ground  before  record- 
ing a subdivision,  but  they  should  be  platted  by  a man  who 
knows  at  least  the  first  principles  of  trigonometry,  and  has  an 
accurately  measured  basis  for  his  work.  ' 

285.  The  Datum-plane. — Levels  referred  to  a permanent 
datum  are  needed  as  soon  as  it  is  apparent  that  the  town  is  to 
be  a living  reality  and  not  simply  a town  on  paper.  The  da- 
tum is  a matter  of  some  importance,  and  should  have  a simple 
relation  to  some  natural  feature  of  the  locality  which  will  re- 
tain a vital  interest  so  long  as  the  town  exists.  There  is  an 
individuality  in  town-sites  which  usually  determines  for  each 
case  very  definitely  what  is  best.  High-water  mark  indicating 
the  danger  of  overflow;  the  lowest  available  outlet  for  a 
drainage  system  in  a flat  country ; the  average  sea-  or  lake- 
level,  as  affecting  commerce ; these  are  often  chosen  and  may 
serve  as  examples.  The  datum  selected  has  its  value  accu- 
rately determined  and  marked  by  a monument  as  enduring 
as  the  granite  hills,  or,  if  that  is  impossible,  as  near  this  stand-' 
ard  as  can  be  secured  ; a block  of  masonry,  with  a single  and 
durable  cap-stone  firmly  bolted  to  its  place,  and  bearing  the 
datum,  or  a known  relation  to  it,  definitely  marked  and  secured 
from  abrasion  is  certainly  possible  for  all. 

286.  The  Location  of  Streets  for  which  the  most  econom- 
ical and  practical  system  of  grades  may  be  secured  is  to  be 
considered  when  the  plat  is  being  prepared.  Grades  are  usu- 
ally established  from  profiles  taken  along  the  centre  lines  of  the 
street  to  be  graded.  This  method  is  direct  and  protects  the 
public  fund,  for  the  grade,  which  can  be  executed  at  minimum 
cost,  the  street  being  considered  by  itself,  can  be  determined 

24 


370 


SURVEYING. 


from  such  a profile.  The  method  fails  from  the  fact  that  it 
treats  the  fund  raised  by  taxation  as  the  sum  total  of  the  pub- 
lic interest.  Parties  representing  abutting  property  appear 
before  the  legislative  body  which  has  final  action  and  seek  to 
amend  the  recommendation  of  the  engineer,  claiming  that  in- 
terests which  should  receive  consideration  are  injured  by  the 
grades  proposed.  It  seems  plain  that  whatever  is  recommend- 
ed by  the  city’s  officer  should  have  the  moral  weight  which 
attaches  to  an  impartial  consideration  of  all  the  interests  which 
the  city  fathers  are  bound  to  recognize.  But  this  involves  a 
change  of  method.  The  contour  map  of  the  district  involved 
seems  to  offer  some  help  toward  a solution.  Methods  by 
which  a rapid  approximation  of  the  amount  of  cut  and  fill  in- 
volved in  any  proposed  grade  may  be  arrived  at  are  discussed 
in  Chapter  XIII.,  on  the  Measurement  of  Volumes. 

287.  Sewer  Systems. — A well-devised  sewer  system 
touches  very  closely  the  public  health.  The  information 
which  is  necessary  in  order  to  act  intelligently  involves,  if 
storm-water  is  to  be  provided  for,  the  area  and  slopes  of  the 
whole  drainage-basin  in  which  lies  the  area  to  be  sewered. 
This  will  enable  a close  approximation  to  be  made  of  the  work 
required  of  the  mains  at  the  point  of  discharge.  Each  sub- 
district involves  its  own  problem.  The  most  economical 
method  of  reaching  every  point  where  drainage  is  necessary 
is  learned  by  studying  the  details  of  topography.  Borings 
along  the  lines  of  proposed  work  to  determine  the  character  of 
the  soil  and  the  depth  of  the  bed-rock  are  necessary  in  order 
that  contractors  may  bid  intelligently.  This  species  of  under- 
ground topography  sometimes  modifies  the  location  fixed  by 
surface  indications. 

288.  Water-supply. — The  need  of  a water-supply  fur- 
nishes new  work  to  the  surveyor.  The  distance  and  elevation 
of  the  source  of  supply,  the  topography  of  the  country  through 
which  aqueducts  or  mains  must  be  brought,  eligible  sites  for 


CITY  SURVEYING. 


371 


reservoirs,  with  their  relation  in  distance  and  elevation  to  all 
points  to  be  supplied,  are  to  be  furnished  to  the  hydraulic 
engineer.  The  datum-plane  for  these  maps  and  that  of  the 
town  should  correspond. 

289.  The  Contour  Map,  which  is  so  generally  useful  from 
the  time  the  town  is  first  planned  until  public  improvements 
cease  to  be  considered,  if  surveyed  carefully  at  first,  has  no 
need  to  be  retraced  each  time  such  a map  is  useful.  It  had 
best  be  drawn  in  sections  of  sufficient  scale  for  a working-plan, 
and  so  arranged  that  when  adjacent  sections  are  placed  side 
by  side  the  contour  lines  will  be  continuous.  If  the  contours 
of  the  natural  surface  are  drawn  in  india-ink,  and  the  contours 
showing  the  changes  made  by  different  kinds  of  public  work 
be  drawn  in  some  color,  the  map  may  give  a great  amount  of 
information  without  becoming  confused. 

METHODS  OF  MEASUREMENT. 

290.  The  Retracing  of  Lines  comes  with  the  private  use 
of  lots  or  blocks  and  with  the  execution  of  public  improve- 
ments. The  demand  for  this  class  of  work  comes  not  once 
only,  but  many  times,  and  never  ceases  while  there  is  life  and 
growth.  The  changes  to  which  these  forces  give  rise  furnish 
the  main  demand  for  knowing  along  what  lines  growth  may 
proceed  unchallenged.  The  man  who  first  fences  a lot  in  the 
middle  of  an  unimproved  block  can  ill  afford  to  risk  being  com- 
pelled to  move  his  fence  for  what  a survey  would  cost.  But 
the  first  attempt  to  go  over  any  part  of  a subdivision  and 
locate  a lot-line  raises  the  question,  how  nearly  alike  can  a 
surveyor  measure  the  same  distance  twice,  or  how  nearly  alike 
can  two  surveyors  measure  the  same  distance.  If  the  distance 
noted  on  the  recorded  plat  was  not  measured  correctly,  the 
resurvey  must  differ  from  it,  or  by  chance  make  a mistake  of 
the  same  amount.  The  difference  which  appears  by  compar- 


372 


SUJ^VEYING. 


ing  results  is  not  the  error  which  exists  in  either  the  original 
or  the  resurvey;  it  may  be  more  than  either  error,  it  may  be 
less,  being  the  algebraic  difference  of  the  two  errors.  If  there 
is  no  difference  it  means  that  the  work  is  uniform,  and  may  be 
correct,  but  both  may  also  be  in  error  a like  amount.  It  has 
happened  in  the  days  of  twenty-foot  rods  and  in  a city  of  con- 
siderable size  that  every  rod  used  by  surveyors  was  too  long. 
The  change  to  steel  tapes  has  not  set  matters  wholly  right. 
If  a man  compares  steel  tapes  bearing  the  brand  of  the  same 
manufacturer  and  offered  for  sale  in  the  same  shop,  he  soon 
ceases  to  be  surprised  at  a very  appreciable  difference  in 
length. 

291.  Erroneous  Standards. — How  long  is  a ten-foot  pole 
or  a hundred-foot  tape  is  a pertinent  and  fundamental  ques- 
tion. It  cannot  be  ignored  when  deeds  call  for  a distance 
from  some  other  point,  as  fixing  the  beginning-point  of  the 
parcel  conveyed.  When  the  deed  describes  lot  number  — , as 
shown  on  the  recorded  plat,  there  is  a theory  in  accordance 
with  which  uniformity  is  all  that  is  required — a distribution  of 
the  distance  between  monuments  in  proportion  to  the  figures 
of  the  record.  Property  is  often  laid  out  with  a view  to  this 
theory  of  surveying.  So  long  as  block-boundaries  are  definitely 
marked,  a degree  of  precision  is  very  readily  secured  by  this 
method  which  is  rarely  attained  when  surveyors  attempt  to 
measure  standard  distances.  If  the  surveyor  faithfully  meas- 
ures the  block  through  and  every  time  distributes  what  he 
finds  in  proportion  to  the  record,  though  his  block  distances 
may  not  agree  with  the  record  or  with  themselves,  the  lot-lines 
will  be  much  more  likely  to  be  the  same  than  if  he  measures 
his  record  distance  and  stops  at  the  lot.  This  method  assumes 
that  the  lots  abut  one  upon  another,  and  reach  from  one  monu- 
ment to  the  other.  But  if  this  be  true,  the  distances  noted 
must  often  refer  to  some  empirical  standard  peculiar  to  this 
block  and  not  to  the  United  States  standard  established  by 


CITY  SURVEYING. 


373 


law.  But  the  courts  recognize  no  standard,  so  far  as  the 
author  knows,  but  that  which  is  established  by  law.  So  that 
when  a surveyor  comes  to  mark  lot  one,  finds  the  corner  of  the 
• block,  and  drives  his  stake  by  measuring  from  it  the  distance 
which  the  record  assigns  to  lot  one,  it  is  hard  to  prove  that  he 
has  not  measured  according  to  the  subdivision,  although  he 
has  given  no  thought  to  the  distance  which  remains  for  the 
other  lots.  But  trouble  begins  right  here,  for  the  theory  which 
is  correct  for  lot  one  cannot  be  very  wrong  for  lot  two ; con- 
tinue the  process  to  lots  six  and  eight,  and  give  to  another  sur- 
veyor who  has  been  doing  the  same  kind  of  work  at  the  other 
end  of  the  block  an  order  to  survey  lot  seven.  A conflict  in 
this  case  is  certain  unless  the  surveyor  who  laid  out  the  sub- 
division, and  each  of  the  others  since,  knew  the  length  of  his 
tape  and  knew  how  to  measure. 

292.  True  Standards. — The  Coast  Survey  Department  of 
the  U.  S.  furnishes  at  small  cost  rods  of  standard  length  at  a 
temperature  which  is  stamped  on  the  rods.  Using  a pair  of 
these  so  as  to  measure  by  contact,  a standard  test-rod  of  any 
desired  length  can  be  laid  off  and  only  such  marks  retained 
as  may  be  desired.  This  test-rod  should  be  of  the  same  ma- 
terial as  the  tape  to  be  tested,  in  order  that  it  may  have  the 
same  coefficient  of  expansion  by  heat  and  may  not  be  affected 
by  humidity  of  the  atmosphere.  Care  being  taken  that  the 
tape  is  of  the  same  temperature  as  the  rod,  be  it  30°,  90°,  or 
60°,  when  the  test  is  made,  then  the  tape  is  correct  at  the  tem- 
perature at  which  the  rod  is  correct,  and  this  is  known  by  the 
U.  S.  stamp,  and  has  no  reference  to  the  temperature  at  the 
time  of  the  test.  In  some  styles  of  tape  the  ring  may  be 
shaped  to  make  the  necessary  adjustment  to  standard  length. 

Where  and  how  to  construct  a standard  rod,  and  how  to 
care  for  it  so  that  it  may  be  permanently  reliable,  each  indi- 
vidual had  best  determine  for  himself.  It  should  be  fastened 
in  its  place  in  such  a manner  that  it  can  expand  and  contract 


374 


SUR  VE  YING. 


freely,  i.e.,  without  any  strain  from  its  supports.  If  it  is  made 
of  separate  parts,  these  should  be  so  joined  together  that  there 
can  be  no  lost  motion  between  the  pieces.  The  whole  requires 
protection  from  the  weather  and  to  be  so  supported  that  it 
cannot  be  bent  by  a blow.  The  writer  has  solved  this  problem 
for  himself  in  the  following  way : Bars  of  tool  steel  one  inch 
wide  and  one  fourth  of  an  inch  thick  are  joined,  as  shown  in 
the  sketch,  to  make  the  desired  length  50  feet  +;  the  whole  is 


Fig.  105. 


fastened  to  the  office  floor  by  screws  which  hold  the  middle 
firmly,  but  each  side  of  the  middle  the  holes  drilled  for  the 
screws  are  slotted  sufficiently  to  allow  for  any  possible  change 
of  temperature.  The  joints  are  so  close  that  a light  blow  is 
necessary  to  bring  the  parts  to  place;  the  screws  were  set 
home  and  then  withdrawn  a little,  so  that  they  should  not 
cause  friction  with  the  floor.  After  the  fastening  was  com- 
pleted the  standard  marks  were  cut  upon  the  rod. 

293.  The  Use  of  the  Tape. — It  is  one  thing  to  have  a 
tape  of  correct  length  ; it  is  another  thing  to  be  able  to  use  it. 
In  an  improved  town  with  curb-lines  clear,  perhaps  the  most 
obvious  method  is  by  a measurement  along  the  grade  with  the 
same  tension  as  that  at  which  the  tape  is  tested.  It  is  then 
necessary  to  correct  for  temperature  and  to  note  all  changes  of 
grade,  reducing  the  observed  distance  on  each  grade  by  the 
versed  sine  of  the  inclination  or  by  the  formula  given  in  Chap. 
XIV.  By  this  method  the  tape  is  supported  for  its  entire  length, 
and  it  is  practicable  to  use  a tape  two  or  three  hundred  feet 
long  to  advantage  provided  there  are  enough  assistants  to 
keep  it  from  being  broken.  A difficulty  arises  in  the  use  of 


CITY  SURVEYING. 


375 


this  method  from  the  fact  that  the  town  is  not  made  for  the 
convenience  of  surveyors,  and  curb-lines  are  not  usually  clear 
where  measurements  are  needed,  but  are  obstructed  by  piles 
of  building  matenal,  bales  of  merchandise,  etc.,  and  in  some 
towns  the  streets  are  so  dirty  that  the  graduation  could  not  be 
seen  long  if  a tape  were  used  in  this  way  ; it  would  also  be  so 
covered  with  drying  mud  that  it  could  not  be  rolled  in  the 
box  when  out  of  use,  hence  would  be  frequently  broken. 
Tapes  that  are  wound  on  a reel,  and  have  no  graduations  to 
speak  of,  could  be  used  in  the  mud,  but  the  other  objections 
mentioned  would  still  make  the  method  of  very  limited  appli- 
cation. It  is  further  to  be  noted  that  the  laying-out  of  the 
town,  which  is  the  basis  of  all  later  work,  has  all  to  be  done 
before  the  streets  are  graded  or  the  curbs  set.  This  work 
must  be  done  by  some  other  method. 

The  usual  method  is  to  keep  the  ends  of  the  tape  horizon- 
tal by  using  a plumb  at  that  end  of  the  tape  where  the  surface 
is  lowest,  and  often  at  both  ends  if  the  ground  is  so  irregular 
or  so  covered  with  brush  and  weeds  that  the  tape  must  be 
kept  off  the  ground.  The  tape  assumes  a curved  form,  and 
the  horizontal  distance  is  something  less  than  the  length  of  the 
tape.  There  is  also  a tension  in  the  tape  which,  on  account  of 
the  elasticity  of  the  metal,  somewhat  increases  its  length.  As 
the  tension  increases  the  sag  diminishes,  hence  there  is  a 
degree  of  tension  such  that  its  effect  is  equal  and  opposite  to 
the  effect  of  the  sag.  Call  this  the  normal  tension.  If  a line  is 
measured  with  a pull  less  than  the  normal  tension  for  the  tape 
used,  the  tape  will  sag  too  much  and  there  will  be  a minus 
error  due  to  this  excessive  sag  ; if  the  pull  used  exceeds  the 
normal  tension,  there  will  be  a plus  error  due  to  this  excess. 
If  the  pull  has  been  uniform  the  total  error  in  either  case  is 
proportional  to  the  length  of  the  line  ; but  if  the  pull  has  not 
been  uniform  the  error  has  varied  irregularly  with  each  length 
of  tape  and  can  most  readily  be  calculated  by  retracing  the  line 


376 


SURVEYING. 


and  using  the  proper  tension.  In  practice  the  tape  is  tested 
with  a known  tension,  and  a tension  so  much  above  the  “ nor- 
mal ” is  adopted  for  field  use  that  its  plus  error  is  equal  to  the 
plus  error  of  the  test. 

294.  To  determine  the  Normal  Tension  in  a tape  sup- 
ported at  given  intervals.  The  tape  forms  a catenary  curve, 
since  it  carries  no  load  but  its  own  weight  and  is  of  uniform 
section. 

Let  P — horizontal  tension  (pull)  ; 

w — weight  of  a unit’s  length  of  tape ; 
e — base  of  Naperian  logarithms  ; 
s — length  of  curve  from  origin  ; 

/ = distance  between  supports  ; 

W = wl  = weight  of  tape  ; 

X and^  = horizontal  and  vertical  coordinates,  origin  at  low- 
est point ; 

X = for  cases  considered. 


Then  by  mechanics,* 


y = 


D wx 

— {e~P-\-e 

2W^  ‘ 


•wx 

~p 


-2), 


and 


We  observe  (i),  that  if  ^ is  constant and  s are  constant  for 
the  same  length  of  tape  ; (2),  if  P be  measured,  say  ten  pounds. 


■*  The  discussion  here  given  is  rigid,  but  both  the  development  and  the  evalu- 
ation of  the  equations  are  laborious.  If  the  curve  be  assumed  to  be  a parabola, 
which  It  may  when  the  sag  is  small,  the  development  is  much  simpler.  See  the 
treatment  of  this  subject  in  Chapter  XIV. — J.  B.  J. 


CITY  SURVEYING. 


377 


as  a working  condition,  j/ and  ^ will  vary  with  the  weight  of 

PI  P . 

every  tape  used,  hence  jp  = —■  is  the  ratio  which  must  be 

constant ; (3),  if  the  surveyor  can  keep  constant,  the  same 
conditions  keep  s constant,  and  if  jy  varies  s must  vary;  (4),  if 

P . 'WX 

x{=  \l')  varies,  and  --  varies  in  the  same  ratio,  then  is  con- 
stant, hence  the  parts  of  the  equations  in  parenthesis  are  con- 

P 

stant  and  y and  s vary  as  I and  — . 

^ ^ w 


TABLES  SHOWING  NORMAL  TENSION  AND  EFFECT  OF 
VARIABLE  TENSION. 


/ = 100  feet. 


JT  = 50  feet. 


Sag. 

Pull. 

P 

w' 

y- 

(2^  - /) 

— error. 

P 

JV' 

Elonga- 

tion 

-f-  error. 

ft. 

ft. 

ft. 

800 

1.56 

0.065 

8 

0.010 

900 

1-39 

0.051 

9 

O.OII 

1000 

1.25 

0.040 

10 

0.012 

1 100 

1. 14 

0.033 

II 

0.014 

1200 

1.04 

0.028 

12 

0.015 

1300 

0.96 

0.023 

13 

0.016 

1400 

0.89 

0.020 

14 

0.017 

1500 

0.83 

0.017 

15 

0.019 

1600 

0.78 

0.014 

16 

0.020 

1800 

0.70 

O.OII 

' 18 

0.022 

2000 

0.62 

0.009 

20 

0.025 

2400 

0.52 

0.007 

24 

0.030 

Resultants  ± 

Error  in  1 

Error  in  1000  ft. 

- 

+ 

- 

+ 

ft. 

0.055 

0.040 

0.028 

0.020 

1 0.013 

0.007 

0.002 

ft. 

ft. 

0.55 

0.40 

0.28 

0.20 

0.13 

0.07 

0.02 

ft. 

0.002 

0.006 

O.OII 

0.016 

0.022 

0.02 

0.06 

0.  II 

0.  16 

0.22 

378 


SURVEYING. 


l — 50'.  X = 25'. 


Sag. 

Puli.. 

Resultants  ± 

P 

y. 

{is  — 1) 

P 

IV' 

Elonga- 

tion 

Error  in  / 

Error  in  looo  ft. 

tv 

-f-  error. 

- 

+ 

- 

ft. 

ft. 

ft. 

400 

0.78 

0.033 

8 

0.003 

0.030 

0.60 

500 

0.63 

0.020 

10 

0.003 

0.017 

0.  34 

600 

0. 52 

0.014 

12 

0.004 

0.010 

0.21 

700 

0.45 

0.010 

14 

0.004 

0.006 

0.  II 

800 

0-39 

0.007 

16 

0.005 

0.002 

0.04 

qoo 

0.35 

0.006 

18 

0.006 

1000 

0.31 

0.004 

20 

0.006 

0.002 

0.03 

1100 

0.28 

0.004 

22 

0.007  ' 

0.003 

0.06 

1200 

0. 26 

0.004. 

24 

0.008  1 

0.004 

0.08 

I'^OO 

0.24 

0.003 

26 

0.008 

0.005 

0.  10 

1400 

0.22 

0.003 

28 

0.009 

0.006 

0.12 

1500 

0.21 

0.002 

30 

0.009 

0.007 

0.14 

1600 

0.  IQ 

0.002 

32 

0.010 

0.008 

0.  16 

1700 

0.18 

0.002 

34 

O.OII 

0.009 



0.18 

1800 

0.17 

0.001 

36 

O.OII 

0.010 

0.20 

Assuminof  values  of  — , the  formulas  are  readily  solved  for 

any  assumed  distance  between  supports  and  the  results  tabu- 
lated ; seven-place  logarithms  are  best  for  this  work. 

The  100'  tape  is  chosen  because  it  furnishes  a ready  means  of 
calculating  a table  for  any  other  length  of  tape  by  a decimal 
reduction  of  the  errors,  per  1000',  in  proportion  to  the  length 

P 

desired,  and  tabulated  with  values  of  — reduced  in  the  same 

w 

proportion.  There  are  those  who  use  the  100'  tape  free-hand, 
with  16  to  20  pounds  pull,  and  say  they  do  the  work  uniformly. 


CITY  SURVEYING. 


379 


In  the  ordinary  formula  for  elongation,  A 


PL 

* 

Ek^ 


we  have 


the  section  k,  a multiple  of  w.  The  foregoing  tables  are  calcu- 
lated from  the  value  w = 3.4^.  The  tension  in  the  tape  P 


*E  is  the  modulus  of  elasticity  in  pounds  to  the  square  inch,  and  ^ is  the 
area  of  the  cross-section  in  square  inches,  Z being  given  in  the  same  denomina- 
tion as  A. 


38o 


SURVEYING. 


differs  from  the  horizontal  tension  P,  so  thatP  = P secant  i 
{i  — inclination  to  the  horizontal),  a second  difference  which 
is  so  small  that  it  may  be  neglected.  Let  E = 27500000  (see 

Chapter  XIV.),  hence  ^ — nearly. 

The  same  facts  for  1000  feet  distance  are  shown  in 
Fig.  106.  In  the  tables  the  plus  and  minus  errors  are  shown 
separately  for  a single  length  of  tape  only,  and  combined 
for  1000'  feet ; in  the  figure  they  are  separated  for  the 
whole  distance  and  the  resultants  of  the  table  are  the  vertical 
intercepts  between  the  curves  (minus  errors)  and  the  straight 
line  (plus  errors).  The  sag  for  a single  length  of  tape  and  cor- 


responding — 


IS 


w 


shown  by  dotted  curved  lines ; these  are 


plotted  to  a reduced  vertical  scale  which  is  shown  at  the  right 
of  the  sketch. 

295.  The  Working  Tension. — In  using  these  tables  it  is 
best  to  measure  the  sag  until  the  necessary  pull  for  the  tape  is 
learned.  When  the  ends  of  the  tape  are  at  a known  elevation 
above  a level  surface,  a rule  at  the  middle  of  the  tape  will 
show  whether  the  pull  is  right.  The  fore  chainman  should 
learn  to  pull  steadily,  not  with  a jerk,  as  he  sticks  the  pin.  A 
more  emphatic  statement  than  the  figure  itself  is  of  the 
worthlessness  of  an  unsteady  hand  at  the  forward  end  of  the 
tape  it  would  be  hard  to  make.  A consciously  constant 
pull,  the  same  every  time,  is  necessary  for  good  work.  To  ob- 
serve the  sag  is  the  surveyor’s  means,  in  the  field,  of  knowing 
that  the  work  is  being  done.  He  soon  learns  to  judge  with 
considerable  accuracy  whether  the  proper  pull  is  constantly 
maintained.  The  proper  pull  is  determined  by  the  tension  at 
which  the  tape  is  tested  ; call  this  p.  Then,  having  weighed 

the  tape,  . Seek  the  plus  error  from  elongation  for  this 


w 


value  of  — ; then  find  the  same  plus  error  between  the  curve  for 
w ^ 


CITY  SURVEYING. 


381 


that  length  of  tape  and  the  straight  line ; the  corresponding  — 
is  right  for  field  use. 

For  example,  a 50'  tape  weighs  six  ounces,  and  the  pull, 
when  tested,  was  five  pounds;  ^ ^ ^ = 666,  and  the 

tV 

elongation  = o'.o83.  The  curve  for  a 50'  tape  marked  — 
error  from  sag  is  distant  from  the  line  marked  -j-  error  from 

P 

pull  the  same  amount  when— - = 1233.  Whence  P=  1233 

X tV  50  — 9i  pounds,  and  the  sag  = o'.25.  When  a tape 
is  to  be  suspended  freely  in  use,  the  tension  at  the  test,/,  should 
not  be  such  that  the  working  tension  Pwill  be  so  great  as  to 
be  impracticable ; but  it  is  also  to  be  noted  that  slight  varia- 
tions of  pull  do  not  affect  the  result  as  much,  when  the  tension 
is  considerably  above  the  normal,  as  the  same  variations  would 
affect  it  if  the  tension  were  at  or  below  the  normal. 

296.  The  Effect  of  Wind.  — A very  moderate  wind  has  a 
marked  effect  on  the  sag  of  the  tape ; the  wind-pressure  on  the 
surface  of  tape  exposed  increases  the  sag  and  gives  it  a diago- 
nal instead  of  a vertical  direction.  The  exposed  surface  of  the 
tape  constantly  changes,  and  this  results  in  vibrations  which 
make  it  difficult  to  tell  where  either  end  of  the  tape  is.  The 
effect  of  its  action,  which  is  a minus  error,  varies  approximately 
as  the  square  of  the  length  of  tape  exposed.  The  effect  of 
winding  up  part  of  the  tape  so  as  to  use  a shorter  length  is  to 
increase  the  use  of  the  plumb,  which  is  also  affected  by  the 
wind,  and  the  result  is  a loss  of  a part  or  all  that  is  gained.  A 
high  working  tension  reduces  the  effect  of  the  wind.  But  the 
only  way  to  eliminate  this  source  of  error  is  to  cease  from  any 
piece  of  work  when  the  wind  is  so  high  that  it  cannot  be  done 
as  it  should  be  done.  There  are  estimates,  topography,  etc., 
which  do  not  require  a high  degree  of  precision  and  which 
can  be  done  when  other  work  cannot. 


382 


SUJ^  VE  YING. 


297.  The  Effect  of  Slope. — When  the  tape  is  used  with 
its  ends  at  different  elevations,  if  it  hangs  freely  its  lowest 
point  would  not  be  in  the  middle,  but  nearer  the  lower  end. 
The  corrections  for  sag  and  pull  still  apply,  however,  with 
inappreciable  error,  for  all  practicable  cases.  The  normal 
tension,  therefore,  remains  the  same  as  for  a level  tape.  A 
correction  must  now  be  made,  however,  for  the  grade,  the 
value  of  which  is  / vers,  f,  where  / is  the  distance  measured 
along  the  slope,  and  i is  the  angle  with  the  horizontal.  The 
measured  distance  is  always  too  great  by  this  amount.* 

The  available  means  by  which  the  tape  may  be  kept  level 
are:  (i)  The  judgment  of  two  field-hands.  (2)  On  difficult 
lines,  the  presence  of  the  surveyor  standing  at  one  side  where 
his  position  has  some  advantages.  A distant  horizon  often  very 
sharply  defines  the  horizontal.  (3)  Where  streets  are  im- 
proved, although  it  may  be  impracticable  to  measure  along  the 
slope,  the  known  fall  per  100  feet  will  give  the  needed  infor- 
mation. (4)  Where  none  of  these  methods  are  sufficient,  test 
the  judgment  by  plumbing  at  different  heights  and  correcting 
the  pin  if  necessary.  These  methods  will  eliminate  the  worst 
errors;  but  where  it  is  necessary  to  measure  lengths  of  five 
or  ten  feet,  and  then  plumb  from  above  the  head,  the  uncor- 
rected remnant  will  be  considerable,  probably  that  due  an 
inclination  of  two  per  cent  on  the  whole  length  of  such 
lines,  with  very  careful  work  to  get  so  near.  This  difference 
in  the  character  of  lines  is  to  be  taken  into  the  account  in 
balancing  the  survey.  Note  that  the  resultant  error  is  always 
minus. 

298.  The  Temperature  Correction. — The  temperature  of 
the  tape  at  the  time  when  the  work  is  done  affects  the  result. 
This  is  not  the  temperature  in  the  shade  that  day,  nor  the 


* This  question  is  fully  discussed  in  Chapter  XIV.,  where  the  correction  is 
found  in  terms  of  the  difference  in  elevation  of  the  two  ends. — J.  B.  J. 


CITY  SURVEYING. 


383 


reading  at  the  nearest  signal  station,  but  is  the  tempera- 
ture out  on  the  line,  under  the  conditions  which  exist  there. 
A grass-covered  slope,  descending  away  from  the  sun,  will 
often  show  at  the  same  time  as  much  as  twenty  or  thirty 
degrees  lower  temperature  than  a bare  hillside  inclining 
toward  the  sun.  The  thermometer  is  needed  with  the  work. 
If  the  co-efficient  of  expansion  is  not  known,  use  0.0000065 
for  1°  F. 

It  is  very  desirable  in  a city-surveyor’s  work  that  he  be  able 
to  apply  his  corrections  at  once  while  in  the  field.  If  he  goes 
out  to  measure  any  given  distance,  he  must  be  able  to  fix  his 
starting-point  and  drive  his  stake  at  the  finish.  If  the  weather 
is  hot  or  cold,  he  knows  what  it  differs  from  the  temperature 
at  which  his  tape  is  tested,  and  applies  the  correction  at  once 
to  the  whole  distance.  He  watches  that  the  pull  is  right, 
that  the  tape  is  kept  horizontal,  that  the  work  stops  when 
the  wind  is  too  severe,  and  that  the  checks  show  the  desired 
accuracy. 

299.  Checks. — Every  piece  of  work  should  be  carried  on 
till  it  checks  upon  other  work,  verifying  its  accuracy  within 
desired  limits.  This  method  ties  up  every  survey  at  both  ends. 
In  order  to  be  prepared  to  do  this  expeditiously,  the  surveyor 
should  lay  out  general  lines  which  should  be  joined  into  a sys- 
tem embracing  the  town-site.  The  lines  of  leading  streets  and 
the  boundary-lines  of  additions  give  most  valuable  information 
when  made  parts  of  such  a system.  This  borders  on  the  geo- 
detic idea,  but  it  will  generally  be  impracticable  to  determine 
the  lengths  of  these  lines  by  triangulation  from  a measured 
base,  for  the  stations  can  very  rarely  be  so  chosen  that  the 
angles  can  be  measured  upon  the  whole  length  of  the  lines,  or 
the  diagonals  be  observed  at  all.  Still,  the  angles  should  be 
measured  upon  the  best  base  practicable.  Permanent  build- 
ings and  existing  monuments  showing  the  lines  of  intersecting 
streets  should  be  noted  both  for  line  and  distance. 


384 


SUR  VE  YING. 


MISCELLANEOUS  PROHLEMS. 

300.  The  Improvement  of  Streets  involves — (i)  The 
estimation  of  the  earthwork  in  the  grading  and  shaping  of  the 
street.  (2)  The  location  of  the  improvements  along  the  lines  of 
the  dedicated  streets.  City  ordinances  usually  prescribe  a cer- 
tain width  of  sidewalks  and  roadway  for  each  width  of  street. 
(3)  The  location  of  improvements  at  the  grade  fixed  by  ordi- 
nance. (4)  The  estimation  of  materials  furnished  by  contractors 
and  used  in  the  work.  The  position  of  monuments  which  will 
be  disturbed  during  the  progress  of  the  work  is  preserved  by 
witness-stakes  driven  beyond  the  limits  of  disturbance.  When 
this  precaution  is  neglected  it  results  in  all  sorts  of  angles  and 
offsets  in  the  curb-lines,  in  cases  where  there  is  surplus  or  defi- 
ciency in  the  original  survey.  Take  a case  improved  one 
block  at  a time,  where  the  first  block  is  established  by  record 
distance  from  the  right,  the  second  block  by  record  distance 
from  the  left,  and  a third  by  running  from  this  last  point  to 
a point  established  at  the  end  of  the  third  block  by  measuring 
again  from  the  right,  etc.  The  resulting  lines  of  curb  will  not 
give  a suggestion  of  where  the  street  was  laid  out.  Some  sur- 
veyors are  accustomed  to  replace  from  their  witness-stakes  the 
monuments  on  the  new  grade.  Such  a practice  is  certainly 
to  be  commended  ; the  small  cost  to  the  public  treasury  can 
well  be  borne  for  the  public  good. 

301.  Permanent  Bench-marks. — In  order  to  secure  accu- 
racy and  uniformity  in  elevations  throughout  a city,  bench- 
marks are  established  by  running  lines  of  levels  radiating  from 
the  directrix,  and  checking  the  work  by  cross-lines  at  conven- 
ient intervals,  these  cutting  the  whole  territory  into  small  par- 
cels, so  that  a standard  bench-mark  will  never  be  far  from  any 
work  which  must  be  done.'^  This  work  is  carried  on  as  far  as 


* These  various  lines  of  levels  will  form  a network,  such  as  that  shown  in 
Chapter  XIV.,  which  should  be  adjusted  once  for  all  as  described  in  that  chapter, 


CITY  SURVEYING. 


385 


grades  are  established,  and  generally  as  far  as  the  city  officers 
are  prepared  to  propose  grades  for  adoption  by  ordinance. 
There  is  a view  of  what  constitutes  or  is  essential  to  accurate 
methods  which  would  make  every  piece  of  work  start  from 
first  principles,  so  that  it  may  not  depend  in  any  way  upon  er- 
rors involved  in  work  previously  done.  But  work  done  on  this 
plan  does  not  have  to  be  extended  very  far  before  the  results 
will  show  plainly  that  there  is  a wide  margin  between  the  uni- 
formity attained  and  the  accuracy  attempted. 

302.  The  Value  of  an  Existing  Monument  is  based  (i)  on 
the  fact  that  it  corresponds  in  character  and  position  to  a mon- 
ument described  on  the  recorded  plat ; (2)  on  the  custom  to 
place  monuments  upon  the  completion  of  a survey,  and  on  the 
supposition  that  this  monument  in  question  was  set  in  pursu- 
ance of  such  custom,  although  no  monuments  are  noted  on 
the  plat ; (3)  on  recognition  by  surveyors  and  owners  of  land 
affected  by  it ; (4)  on  the  knowledge  that  it  was  placed  by  a 
competent  surveyor  at  a time  when  data  were  accessible  which 
are  not  now  in  existence.  The  value  of  the  evidence  which 
establishes  or  tends  to  establish  the  reliability  of  the  monument 
is  primarily  a question  for  the  judgment  of  the  surveyor.  His 
decision  must  be  reviewed  and  defended  before  courts  and  ju- 
ries when  there  is  a difference  of  opinion. 

The  monument  is  valueless,  or  less  valuable  in  all  degrees, 
when  there  is  evidence  that  it  has  been  disturbed.  It  some- 
times happens  that  there  is  no  better  way  to  establish  a corner 
than  to  straighten  up  a stone  which  is  leaning,  but  has  not 
been  thrown  entirely  out  of  the  ground.  Inquiry  often  brings 
out  the  fact  that  a stone,  after  being  completely  out  of  the 
ground,  has  been  reset  either  by  agreement  of  owners  adja- 


:ind  so  one  elevation  obtained  for  each  bench-mark.  It  is  common  for  each 
bench-mark  in  a city  to  have  numerous  elevations  differing  by  several  tenths  of 
a foot,  and  all  of  about  equal  credence. — J.  B.  J. 

25 


386 


S UR  VE  YING. 


cent,  or  by  the  reckless  individual  who  did  the  mischief,  and  is 
still  pointed  out  as  the  stone  the  surveyor  set.  As  a recog- 
nized corner  such  a stone  has  some  value,  i.e.,  it  is  to  be  sup- 
posed that  it  is  somewhere  in  the  right  neighborhood;  but  if  its 
position  can  be  verified  from  other  points  which  have  not  been 
disturbed  the  work  should  be  retraced.  If  the  original  survey 
was  made  in  a careless  way  or  the  corner-stones  were  badly 
set,  they  may  help  a careful  man  to  come  to  an  average  line 
which  shall  correspond  with  the  recorded  plat.  Monuments 
are  sometimes  moved  or  destroyed  maliciously.  It  is  wise  for 
a surveyor  to  test  discreetly  everywhere,  but  to  be  especially 
careful  where  there  has  been  quarrelling  about  lines. 

There  is  a principle,  recognized  to  some  extent  by  the 
courts,  that  the  existing  monument  is  the  evidence  of  the  orig- 
inal survey,  whether  or  not  it  is  called  for  by  the  recorded 
plat.  The  custom  that  the  surveyor  making  the  subdivision 
and  the  plat  for  record  shall  set  corner-stones  is  so  far  fol- 
lowed that  this  is  generally  true,  cases  of  accident,  carelessness, 
and  mischief,  and  such  cases  as  that  mentioned  below,  being 
somewhat  exceptional,  but  many  times  very  real.  It  is  some- 
times attempted  to  go  a step  further  and  affirm  that  the  re- 
corded plat  is  the  record  of  the  survey.  This  reverses  the  or- 
der of  events  in  most  cases,  the  survey  being  made  in  order  to 
mark  upon  the  ground  the  chief  points  of  apian  already  fixed 
upon ; and  as  to  all  the  main  lines,  the  plat  is  not  altered,  how- 
ever carelessly  the  survey  may  be  made.  There  are  subdivisions 
where  no  monuments  were  set  and  where  no  certain  evidence 
is  in  existence  of  how  or  where  the  original  survey  was  made, 
or  whether  any  survey  was  made  at  all,  and  yet  there  is  a re- 
corded plat.  A surveyor  being  called  upon  to  make  a survey 
of  some  parcel  in  such  a subdivision,  sets  stones  in  order  to  se- 
cure recognition  for  his  theory  of  the  proper  location.  If  he 
does  his  work  carefully  he  undoubtedly  does  the  public  a ser- 
vice. Can  any  amount  of  ignorance  of  when  or  why  these 


CITY  SURVEYING. 


387 


stones  were  set  ever  make  them  evidence  of  the  original  sur- 
vey? In  other  cases  some  monuments  maybe  in  existence, 
but  more  would  be  convenient, — points  are  determined  from 
existing  monuments  in  accordance  with  the  recorded  plat  and 
stones  are  set.  Another  surveyor  may  feel  a little  nervous 
about  manufacturing  this  sort  of  evidence  of  the  original  sur- 
vey, or  more  likely,  may  think  it  too  much  trouble  and  a dam- 
age to  the  business,  for  the  more  doubt  the  more  work  for  the 
surveyor,  so  drives  his  stake.  Then  comes  the  owner  who, 
desiring  to  secure  a permanent  corner,  digs  a hole  about  the 
stake  without  taking  offsets,  throws  it  out,  and  sets  in  a stone 
— an  existing  monument!  This  is  no  fancy  sketch,  nor  are 
such  facts  so  very  rare.  The  young  man  who  thinks  he  would 
like  to  be  a surveyor,  but  has  no  eyes  nor  ears  for  facts  like 
these,  had  better  turn  his  attention  to  some  other  business. 
Surveying  is  an  art — not  an  exact  science.* 

303.  The  Significance  of  Possession. — Possession  has  a 
value  in  reestablishing  old  lines  where  all  monuments  have 
disappeared.  It  is  a species  of  perpetuating  testimony  of  their 
positions.  The  average  of  a series  of  improvements  will  often 
give  a very  close  determination  of  where  the  corner  must  have 
stood.  The  practised  eye  accustomed  to  sharply  defined  lines, 
every  lot  having  very  nearly  its  right  quantity,  which  are  cus- 
tomary where  lines  are  well  established,  will  notice  at  once  the 
irregular  possession, — gaps  between  houses,  vacant  spaces 
between  fences  and  houses,  too  little  for  use,  too  much  for 
ornament,  which  may  be  seen  where  lines  are  in  doubt  and 
every  man  expects  the  next  surveyor  to  make  a conflicting 
survey.  Like  the  men  of  the  present,  most  men  in  the  past 
have  preferred  to  be  right — have  made  efforts  to  be  right — 
have  employed  surveyors ; we  can  judge  where  these  men  in 


* Consult  Judge  Cooley’s  paper  on  the  Judicial  Functions  of  the  Surveyor, 
Appendix  A. 


388 


SUR  VE  VING. 


the  past  worked  from  by  seeing  where  their  works  arc.  The 
legal  principle  has  a bearing  here,  that  “ he  who  would  sue  to 
dispossess  another  must  first  show  a better  title.”  Tlic  sur- 
veyor who  attempts  to  dispute  possession  must  show  better 
evidence  than  possession  of  the  right  location  of  the  lines  he 
is  employed  to  retrace. 

304.  Disturbed  Corners  and  Inconsistent  Plats. — The 

work  of  testing  a corner  that  probably  has  been  disturbed  has 
many  points  of  likeness  to  the  work  of  reestablishing  corners 
that  have  disappeared  altogether.  The  recorded  plat  is  in  all 
cases  the  basis  of  the  work.  When  it  records  the  results  of  a 
survey  it  is  to  be  presumed  that  the  surveyor  endeavored  to 
do  accurate  work  ; hence  his  work,  if  not  absolutely  correct, 
was  probably  uniform.  Lines  which  are  shown  by  the  plat 
as  straight  lines  are  to  be  retraced  as  straight  lines.  Lines  in- 
volve less  liability  to  error  than  measurements,  and  are  first  to 
be  considered.  Determine  as  many  points  as  possible  by 
straight  lines  between  existing  monuments.  Then  test  the 
measurements  along  the  extreme  lines  and  the  streets  which 
are  the  basis  of  the  subdivision.  If  the  measurements  between 
undoubted  corners  agree  with  the  plat  so  closely,  or  if  they 
differ  so  uniformly  that  the  presumption  of  accurate  work 
is  justified,  corners  that  are  out  of  line  or  out  of  proportionate 
distance  have  the  burden  of  proof  against  them.  He  who 
would  claim  for  them  authority  must  show  that  they  have  not 
been  disturbed,  and  that  they  are  consistent  with  some  ra- 
tional location.  If  there  was  no  original  survey,  that  fact  is  no 
excuse  for  careless  Avork  at  a later  time  ; there  is  always  some 
place  to  begin.  The  case  when  the  recorded  plat  does  not 
agree  with  itself  presents  more  difficulties,  such  as  the  follow- 
ing: (i)  The  lines  do  not  give  the  same  points  as  the  distance ; 
(2)  The  distances  disagree  among  themselves;  (3)  The  monu- 
ments disagree  with  both  lines  and  distances  impartially,  or 
agree  with  one  and  disagree  with  the  other,  while  the  general 


CITY  SURVEYING. 


389 


character  of  the  work  negatives  the  supposition  that  they  were 
ever  carefully  set.  The  object  to  be  sought  is  not  to  perpet- 
uate forever  the  blunders  of  the  original  survey,  but  to  seek 
the  most  rational  adjustment  of  all  the  evidence,  so  that  all  parts 
may  be  located  with  a minimum  of  conflict,  and  so  that  no 
one  shall  be  able  to  prove  your  survey  wrong,  i.e.,  show  a 
more  reasonable  location  for  any  part.  A consultation  of 
surveyors  before  too  many  conflicting  interests  have  developed 
is  often  advantageous. 

305.  Treatment  of  Surplus  and  Deficiency. — It  is  gen- 
erally a simpler  problem  to  determine  in  which  block  differences 
of  measurement,  whether  surplus  or  deficiency,  belong  than  it 
is  to  know  what  to  do  with  them  in  the  matter  of  lot-location. 
There  has  never  been  any  theory  invented  for  the  treatment 
of  either  surplus  or  deficiency  which  is  able  to  stand  the  test 
of  the  courts  against  all  combinations  of  circumstances.  A 
few  suggestions  with  the  more  probable  limitations  are  all  the 
help  that  can  be  offered:  every  case  must  be  investigated  for 
itself,  (i)  A distribution  of  the  whole  front  in  proportion  to 
the  record  distances  meets  general  approval,  at  least  in  cases  of 
surplus,  until  it  comes  in  conflict  with  possession.  This  is  just 
the  time  when  an  owner  of  ground  wants  to  know  what  his 
rights  are,  and  it  is  also  the  time  when  no  surveyor  can  tell 
him.  A compromise,  or  the  verdict  of  a petit  jury,  which 
passes  foreknowledge,  are  the  chief  alternatives.  The  ^courts 
say  that  he  who  would  sue  for  possession  must  show  a better 
title.  An  examination  shows  that  each  has  a better  title 
than  any  other  to  so  much  ground  as  the  plat  assigns  to  his 
lot,  but  that  no  one  has  a better  title  than  any  other  one  to 
any  part  of  the  surplus.  The  surveyor  does  wisely  to  take 
note  of  possession  and  make,  if  he  can,  such  a location  as  is  in 
accordance  with  the  record,  and  yet  not  in  conflict  with  posses- 
sion. When  this  is  not  possible,  let  the  map  and  certificate  of 
survey  be  made  in  such  a way  that  they  are  simply  a state- 


SURVEYING, 


390 


merit  of  the  facts.  It  is  not  a surveyor’s  business  to  decide 
legal  questions  or  give  judgment  in  ejectment.  (2)  Because 
a suit  for  surplus  will  not  lie,  it  has  been  thought  that  he  who 
first  took  possession  of  the  surplus  would  be  secure  if  he  were 
only  careful  to  take  it  so  that  every  other  one  might  have  his 
ground.  Trouble  with  this  view  arises  because  it  is  not  possi- 
ble to  locate  the  surplus.  When  one  man  has  appropriated  all 
there  is  in  the  block,  and  the  rest  but  one  have  appropriated 
each  his  proportionate  share,  then  comes  the  last  man.  The 
more  surplus  in  the  block  the  more  he  is  deficient ; he  wants  his 
ground,  and  he  finds  it  easier  to  sue  the  one  man  than  the  twenty. 
Perhaps,  in  order  to  be  sure  of  a case,  he  had  better  sue  them  all. 
The  cases  which  arise  in  practice  take  on  an  infinite  variety  of 
complications  and  are  not  usually  so  simple  as  these  described. 
(3)  The  fact  is,  that  the  idea  that  a subdivision  ought  to 
have  a little  surplus  is  irrational.  The  work  should  be  so 
close  to  the  standard  that  the  surveyor  who  retraces  the  lines 
would  testify : “ According  to  the  best  of  my  knowledge  and 
belief,  there  is  neither  surplus  nor  deficiency  there.  In  retrac- 
ing my  own  work,  which  is  carefully  executed,  I observe  as 
great  discrepancies  as  any  which  I find  in  this  subdivision,  and 
I conclude  that  the  small  difference  which  I observed  in  this 
case  was  as  likely  to  have  been  an  error  in  my  own  work  as  to 
attach  to  the  subdivision.”  (4)  Deficiency  would  seem  to  be 
easier  to  deal  with  than  surplus;  for  when  the  last  man  has  not 
his  ground  he  has  a valid  claim  against  the  original  owner  for  a 
rebate  on  the  purchase-price.  But  the  burden  of  the  difficulty 
in  this  case  falls  on  the  surveyor.  When  a man  brings  his 
deed  and  asks  a survey  of  lot  9,  while  8 and  10  are  unsold  and 
lots  I to  7 are  already  in  possession,  he  leaves  lot  8 its  ground 
and  the  deficiency  in  lot  10.  Suppose  it  turns  out  that  lot  10  is 
next  sold,  and  that  the  surveyor  reports  it  deficient,  the  seller, 
when  waited  on,  may  reply,  “ I have  not  sold  more  ground  in  the 
block  than  I owned ; the  surveyor  has  made  a mistake  in  locat- 


CITY  SURVEYING. 


391 


ing  lot  9.”  This  liability  attaches  to  every  location  which  is 
made  before  every  lot,  between  the  one  located  and  one  corner 
of  the  block,  is  sold.  (5)  It  is  practicable  for  the  original 
owner  to  so  write  his  deeds  as  to  locate  surplus  or  deficiency. 
By  beginning  all  deeds  at  the  record  distance  from  one  street 
and  continuing  this  uniformly  through  the  block,  the  differ- 
ence goes  in  the  lot  farthest  from  the  starting-point ; or  he 
may  continue  the  process  up  to  any  line  which  he  may  choose, 
and  work  from  the  other  end  of  the  block  in  deeding  the  re- 
maining lots ; then  the  difference  falls  upon  the  line  chosen 
and  falls  to  the  share  of  the  lot  abutting  upon  that  line  which 
is  last  deeded.  But  to  approve  this  method  is  to  affirm  the 
practicability  of  absolute  accuracy  in  work.  No  one  can  tell 
how  small  a difference  may  cause  trouble. 

306.  The  Investigation  and  Interpretation  of  Deeds  for 
the  use  of  the  land-surveyor,  dealing  with  the  harmony  or 
conflict  of  the  descriptions,  is  entirely  a different  work  from 
that  of  the  investigator  of  titles,  which  deals  with  the  legal 
completeness  of  the  conveyance.  In  the  older  parts  of  a town 
the  deed  of  the  present  proprietor  frequently  does  not  give 
information  sufficient  to  fix  the  correct  location.  The  key 
may  lie  in  some  boundary  in  an  early  deed  referring  to  a still 
earlier  conveyance  of  adjacent  property.  Or  the  earlier  deeds 
may  give  clearly  defined  locations,  while  the  latter  ones 
say  “ more  or  less”  at  every  point.  In  some  cases  the  deeds 
are  in  such  a condition  that  it  is  impossible  to  tell  what  they 
mean  until  it  is  known  what  the  possession  is.  Skill  in  this 
work  can  only  come  after  considerable  experience  ; local  prac- 
tices must  largely  determine  what  is  necessary. 

307.  Office  Records. — The  surveyor’s  office  when  well 
planned  is  so  arranged  that  no  item  of  information  which 
promises  to  be  useful  shall  be  lost.  The  customary  methods 
of  indexing,  and  of  block-plats  for  keeping  notes,  do  not  take 
a very  firm  hold  on  general  lines  or  the  connections  between 


392 


SUR  VE  YING. 


subdivisions;  they  fail,  in  fact,  in  that  part  of  the  work  wliicli 
has  the  most  vital  relation  to  efforts  at  future  improvement. 
It  is  advisable  to  add  to  the  block-plats  and  indexes  a general 
atlas  of  the  whole  town  for  office  use,  at  a scale  of  say  lOo' 
to  the  inch,  so  that  an  area  nearly  half  a mile  square  may 
appear  on  the  open  pages.  Such  an  atlas  may  show  the  notes 
of  the  general  lines  and  their  angles,  the  base-line  measure- 
ments, the  relation  of  subdivisions  to  one  another,  and  a 
variety  of  other  information  which  it  is  difficult  to  pick  out  in 
the  widely  scattered  field-notes  which  first  gathered  the  in- 
formation, and  which,  with  their  larger  scale,  the  block-plats 
are  not  well  adapted  to  show  in  a connected  form. 

There  are  filed  in  connection  with  deeds  many  plats  which 
do  not  appear  on  the  record  plat-books  of  the  recorder’s 
office  ; these  need  to  be  indexed,  or,  better,  abstracted  for 
office  use. 

The  field-notes,  when  prepared  for  the  surveyor’s  use  in 
the  field,  should  show  in  an  accessible  and  portable  form  all 
the  information  which  the  office  contains  and  which  is  rele- 
vant to  the  survey  in  hand.  Labor  spent  beforehand  in  a 
thorough  preparation  of  accessible  information  is  labor  saved. 

308.  The  Preservation  of  Lines  after  the  monuments 
have  disappeared  is  accomplished  by  means  of  notes  on  build- 
ings, marks  and  notes  on  curbing,  paving,  fences,  etc.  Notes 
on  buildings  describe  not  only  the  character  of  the  building, 
but  the  particular  part  noted,  so  that  another  man,  years 
afterward,  using  the  same  note  would  have  no  doubt  of  the 
identity  of  the  part.  In  a growing  town  the  work  of  keeping 
up  the  notes  goes  on  without  ceasing, — buildings  are  remod- 
elled or  rebuilt,  streets  reconstructed,  destroying  old  marks. 
The  old  becomes  the  new  so  constantly  that  the  surveyor 
who  would  preserve  the  information  which  he  already  has 
must  be  constantly  employed  at  the  work  of  renewal.  There 
is  no  place  either  in  the  street  or  out  of  it  where  the  surveyor 


CITY  SURVEYING. 


393 


can  place  his  mark  and  say  to  all  comers,  “ Touch  not.”  It 
follows  that  whenever  it  is  necessary  to  use  any  mark,  about 
the  permanence  of  which  there  can  be  a shadow  of  a doubt, 
the  permanence  of  the  mark  must  be  shown  by  some  prac- 
ticable test ; it  is  careless  to  assume  it. 

309.  The  Want  of  Agreement  between  Surveyors  arises 
from  differences  of  information  or  of  judgment,  and  in  a less 
degree  from  differences  of  skill.  These  are  all  just  as  human 
elements  as  the  lawyer  deals  with  in  his  work.  Testimony  is 
affected  by  the  interests  of  those  who  speak,  and  the  judg- 
ment varies  with  the  temperament  of  the  individual.  Per- 
haps one  of  the  most  difficult  lines  for  a surveyor  to  draw  is 
that  which  separates  his  confidence  in  his  own  skill  in  retrac- 
ing a survey  which  was  confessedly  inaccurate,  from  his  re- 
liance on  testimony  which  is  evidently  biassed  as  to  the  posi- 
tion or  disturbance  of  monuments,  and  other  facts  which 
may  help  him  to  form  a correct  judgment.  Errors  in  execu- 
tion may  be  kept  within  such  limits  that  work  which  shows 
differences  in  closing  of  i in  5000  should  be  retraced,  and  the 
average  observed  differences  in  one  surveying  party’s  work 
will  not  exceed  i in  20000.  Two  sets  of  men  working  to 
reach  the  same  standard  may  err  in  opposite  directions,  so 
that  differences  between  two  surveyors  may  reasonably  be 
expected  to  be  somewhat  larger  than  either  would  tolerate  in 
his  own  work. 


CHAPTER  XIIL 

THE  MEASUREMENT  OF  VOLUMES. 

310.  Proposition. — The  volume  of  any  doubly-truncated 
prism  or  cylinder^  bounded  by  pla7ie  ends,  is  equal  to  the  area  of  a 
right  section  into  the  length  of  the  element  through  the  centres  of 
gravity  of  the  bases,  or  it  is  equal  to  the  area  of  either  base  into 
the  altitude  of  the  element  joining  the  ceyitres  of  gravity  of  the 
bases,  measured  perpe^idicidar  to  that  base. 

Let  ABCD,  Fig.  107,  be  a cylinder,  cut  by  the  planes  OC 
and  OB,  the  unsymmetrical  right  section  EF  being  shown  in 
plan  in  E' F . Whatever  position  the  cutting  planes  may  have, 
if  they  are  not  parallel  they  will  intersect  in  a line.  This  line 
of  intersection  may  be  taken  perpendicular  to  the  paper,  and 
the  body  would  then  appear  as  shown  in  the  figure,  the  line 
of  intersection  of  the  cutting  planes  being  projected  at  O. 

Let  A — area  of  the  right  section  ; 

A A — any  very  small  portion  of  this  area: 

X — distance  of  any  element  from  O ; 
then  ax  = height  of  any  element  at  a distance  x from  O. 

An  elementary  volume  would  then  be  axAA,  and  the  total 
volume  of  the  solid  would  be  'EaxAA. 

Again,  the  total  volume  is  equal  to  the  mean  or  average 
height  of  all  the  elementary  volumes  multiplied  by  the  area 
of  the  right  section. 

The  mean  height  of  the  elementary  volumes  is,  therefore, 


THE  MEASUREMENT  OF  VOLUMES. 


395 


^axAA 


. But  is  the  distance  from  O to  the 


centre  of  gravity,  of  the  right  section,*  and  a times  this  dis- 
tance is  the  height  of  the  element  LK  through  this  point. 
Therefore,  the  mean  height  is  the  height  through  the  centre  of 


r., 


gravity  of  the  base,  and  this  into  the  area  of  the  right  section 
is  the  volume  of  the  truncated  prism  or  cylinder.  The  truth 
of  the  alternative  proposition  can  now  readily  be  shown. 

Corollary.  When  the  cylinder  or  prism  has  a symmetrical 
cross-section,  the  centre  of  gravity  of  the  base  is  at  the  centre 
of  the  figure,  and  the  length  of  the  line  joining  these  centres 
is  the  mean  of  any  number  of  symmetrically  chosen  exterior 
elements.  For  instance,  if  the  right  section  of  the  prism  be  a 
regular  polygon,  the  height  of  the  centre  element  is  the  mean 
of  the  length  of  all  the  edges.  This  also  holds  true  for  paral- 
lelograms, and  hence  for  rectangles.  Here  the  centres  of  gravity 


* This  is  shown  in  mechanics,  and  the  student  may  have  to  take  it  for 
granted  temporarily. 


SUA'  VE  YANG. 


396 


of  the  bases  lie  at  the  intersections  of  the  diagonals ; and  since 
these  bisect  each  other,  the  length  of  the  line  joining  the  in- 
tersections is  the  mean  of  the  lengths  of  the  four  edges.  The 
same  is  t^rue  of  triangular  cross-sections. 

31 1.  Grading  over  Extended  Surfaces. — Lay  out  the 
area  in  equal  rectangles  of  such  a size  that  the  surfaces  of  the 
several  rectangles  may  be  considered  planes.  For  common 
rolling  ground  these  rectangles  should  not  be  over  fifty  feet 
on  a side.  Let  Fig.  108  represent  such  an  area.  Drive  pegs  at 


1222221 


the  corners,  and  find  the  elevation  of  the  ground  at  each  in- 
tersection by  means  of  a level,  reading  to  the  nearest  tenth  of 
a foot,  and  referring  the  elevations  to  some  datum-plane  below 
the  surface  after  it  is  graded.  When  the  grading  is  completed, 
relocate  the  intersections  from  witness-points  that  were  placed 
outside  the  limits  of  grading,  and  again  find  the  elevations  at 
these  points.  The  several  differences  are  the  depths  of  excava- 
tion (or  fill)  at  the  corresponding  corners.  The  contents  of 
any  partial  volume  is  the  mean  of  the  four  corner  heights  into 
the  area  of  its  cross-section.  But  since  the  rectangular  areas 
were  made  equal,  and  since  each  corner  height  will  be  used  as 
many  times  as  there  are  rectangles  joining  at  that  corner,  we 
have,  in  cubic  yards. 


4 X 27 


[2/..  + 2^’4  + 32/^,  + 4i'/«J.  . . (I) 


THE  MEASUREMENT  OF  VOLUMES. 


397 


The  subscripts  denote  the  number  of  adjoining  rectangles 
the  area  of  each  of  which  is  A. 

From  this  equation  we  may  frame  a 

Rule. — Take  each  corner  height  as  many  times  as  there 
are  partial  areas  adjoining  it,  add  them  all  together,  and  mul- 
tiply by  one  fourth  of  the  area  of  a single  rectangle.  Tnis 
gives  the  volume  in  cubic  feet.  To  obtain  it  in  cubic  yards, 
divide  by  twenty-seven. 

If  the  ground  be  laid  out  in  rectangles,  30  feet  by  36  feet, 

then — — — ■ = = 10;  and  if  the  elevations  be  taken  to 

4 X 27  108 

the  nearest  tenth  of  a foot,  then  the  sum  of  the  multiplied 
corner  heights,  with  the  decimal  point  omitted,  is  at  once  the 
the  amount  of  earthwork  in  cubic  yards.  This  is  a common 
way  of  doing  this  work.  In  borrow-pits,  for  which  this  method 
is  peculiarly  fitted,  the  elementary  areas  would  usually  be 
smaller. 

In  general,  on  rolling  ground,  a plane  cannot  be  passed 
through  the  four  corner  heights.  We  may,  however,  pass  a 
plane  through  any  three  points,  and  so  with  four  given  points 


1 3 3 3 3 4 1 


on  a surface  either  diagonal  may  be  drawn,  which  with  the 
bounding  lines  makes  two  surfaces.  If  the  ground  is  quite 
irregular,  or  if  the  rectangles  are  taken  pretty  large,  the  sur- 
veyor may  note  on  the  ground  which  diagonal  would  most 


398 


SURVEYING. 


nearly  fit  the  surface.  Let  these  be  sketched  in  as  shown  in 
Fig.  109.  Each  rectangular  area  then  becomes  two  triangles, 
and  when  computed  as  triangular  prisms,  each  corner  height 
at  the  end  of  a diagonal  is  used  twice,  while  the  two  other 
corner  heights  are  used  but  once.  That  is,  twice  as  much 
weight  is  given  to  the  corner  heights  on  the  diagonals  as  to 
the  others.  In  Fig.  109,  the  same  area  as  that  in  Fig.  108  is 
A,  shown  with  the  diagonals  drawn  which  best  fit 
the  surface  of  the  ground.  The  numbers  at 
the  corners  indicate  how  many  times  each 
height  is  to  be  used.  It  will  be  seen  that 
each  height  is  used  as  many  times  as  there  are 
triangles  meeting  at  that  corner.  To  derive 
the  formula  for  this  case,  take  a single  rectangle,  as  in  Fig. 
no,  with  the  diagonal  joining  corners  2 and  4.  Let  A be  the 
area  of  the  rectangle.  Then  from  the  corollary,  p.  395,  we 
have  for  the  volume  of  the  rectangular  prism,  in  cubic  yards, 


_ ^ i 7^1  -|-  I -h  K + ^4\ 

~ 2 3 3 ] 


For  an  assemblage  of  such  rectangular  prisms  as  shown  in 
Fig.  109,  the  diagonals  being  drawn,  we  have,  in  cubic  yards, 

y — ^ 

+ ; ...  (3) 

where  A is  the  area  of  one  rectangle,  and  the  subscripts  denote 
the  number  of  triangles  meeting  at  a corner. 


THE  MEASUREMENT  OF  VOLUMES. 


399 


As  a check  on  the  numbering  of  the  corners,  Fig.  109,  add 
them  all  together  and  divide  by  six.  The  result  should  be 
the  number  of  rectangles  in  the  figure.  In  this  case,  if  the 
rectangles  be  taken  36  feet  by  45  feet,  or,  better,  40  feet  by  40.5 
feet,  then  the  sum  of  ,the  multiplied  heights  with  the  decimal 
point  omitted  is  the  number  of  cubic  yards  of  earthwork,  the 
corner  heights  having  been  taken  out  to  tenths  of  a foot. 

The  method  by  diagonals  is  more  accurate  than  that  by 
rectangles  simply,  the  dimensions  being  the  same ; or,  for 
equal  degrees  of  exactness  larger  rectangles  may  be  used  with 
diagonals  than  without  them,  and  hence  the  work  materially 
reduced.  In  any  case  some  degree  of  approximation  is  neces- 
sary. 

312.  Approximate  Estimates  by  means  of  Contours. — 

(A)  Whenever  an  extended  surface  of  irregular  outline  is  to 
be  graded  down,  or  filled  up  to  a given  plane  (not  a warped  or 
curved  surface),  a near  approximation  to  the  amount  of  cut  or 
fill  may  be  made  from  the  contour  lines.  In  Fig.  in  the  full 
curved  lines  are  contours,  showing  the  original  surface  of  the 
ground.  Every  fifth  one  is  numbered,  and  these  were  the  con- 
tours shown  on  the  original  plat.  Intermediate  contours  one 
foot  apart  have  been  interpolated  for  the  purpose  of  making 
this  estimate.  The  figures  around  the  outside  of  the  bound- 
ing lines  give  the  elevations  of  those  points  after  it  is  graded 
down.  The  straight  lines  join  points  of  equal  elevation  after 
grading;  and  since  this  surface  is  to  be  a plane  these  lines  are 
surface  or  contour  lines  after  grading.  Wherever  these  two 
sets  of  contour  lines  intersect,  the  difference  of  their  elevations 
is  the  depth  of  cut  or  fill  at  that  point.  If  now  we  join  the 
points  of  equal  cut  or  fill  (in  this  case  it  is  all  in  cut),  we  ob- 
tain a new  set  of  curves,  shown  in  the  figure  by  dotted  lines, 
which  may  be  used  for  estimating  the  amount  of  earthwork. 
The  dotted  boundaries  are  the  traces  on  the  natural  surface  of 
planes  parallel  to  the  final  graded  surface  which  are  uniformly 


400 


SUf!  VE  YING. 


spaced  one  foot  apart.  These  areas  are  measured  by  the 
planimeter  and  called^,,  etc.  Each  area  is  bounded  by 

the  dotted  line  and  the  bounding  lines  of  the  figure,  since  on 
these  bounding  lines  all  the  projections  of  all  the  traces  unite, 


the  slope  here  being  vertical.  For  any  two  adjoining  layers 
we  have,  by  the  prismoidal  formula*  as  well  as  by  Simpson’s 
one-third  rule, 


3 — 2 

where  h is  the  common  perpendicular  distance  between  the 
sections. 


* For  the  demonstration  of  the  prismoidal  formula  see  p.  403. 


THE  MEASUREMENT  OF  VOLUMES. 


401 


For  the  next  two  layers  we  would  have,  similarly, 


(2) 


or  for  any  even  number  of  layers  we  would  have,  in  cubic 
yards, 


where  n is  an  odd  number,  h and  A being  in  feet  and  square 
feet  respectively. 

{E)  Whenever  the  final  surface  is  not  to  be  a plane  but  a 
surface  which  may  be  defined  by  drawing  the  contour  lines  as 
they  are  to  be  when  the  grading  is  completed,  the  above 
method  may  still  be  used.  Thus,  suppose  a given  tract  of 
ground,  the  contours  of  which  have  been  carefully  determined, 
is  to  be  transformed  into  certain  new  outlines,  as  is  often 
required  in  landscape-gardening  and  in  the  making  of  parks 
and  cemeteries,  the  new  contours  may  be  traced  on  the  plat 
containing  the  original  contours  by  using  a different-colored 
ink.  The  second  set  of  contours  are  now  curved  instead  of 
straight,  as  was  the  case  in  the  preceding  example.  Otherwise 
there  is  no  difference  in  the  methods.  The  intersections  of  the 
two  sets  of  contours  are  marked  with  the  number  of  feet  of 
cut  or  fill,  the  same  as  before,  the  cuts  being  designated  by  a 
plus  and  the  fills  by  a minus  sign.  The  curves  of  equal  cut  or 
fill  are  now  drawn,  preferably  in  an  ink  of  a different  color  from 
the  other  two,  and  areas  measured  and  the  volume  computed 
exactly  as  in  the  former  case.  It  would  also  be  well  to  desig- 
nate the  cut  and  the  fill  curves  by  ink  of  different  shades  but 
of  the  same  color. 

When  a rectangular  area,  as  a city  block,  is  to  be  graded  to 
26 


402 


SU/^  VE  y INC. 


a warped  surface,  which  it  generally  is,  the  contours  of  this 
surface  are  readily  obtained  from  the  street-grades,  and  the 
above  method  used.  For  accurate  measurements,  such  as 
should  be  made  the  basis  of  payment,  the  area  should  be  di- 
vided into  rectangles,  as  previously  described.  These  approxi- 
mate methods  serve  well  for  preliminary  estimates.  They 
may  be  found  useful  in  determining  street-grades  when  it  is 
desired  to  equalize  the  cuts  and  fills  over  the  blocks  rather 
than  on  the  street-lines. 

313.  The  Prismoid  is  a solid  having  parallel  end  areas, 
and  may  be  composed  of  any  combination  of  prisms,  cylinders, 
wedges,  pyramids,  or  cones  or  frustums  of  the  same,  whose 
bases  and  apices  lie  in  the  end  areas.  It  may  otherwise  be 
defined  as  a volume  generated  by  a right-line  generatrix  mov- 
ing on  the  bounding  lines  of  two  closed  figures  of  any  shapes 
which  lie  in  parallel  planes  as  directrices,  the  generatrix  not 
necessarily  moving  parallel  to  a plane  director.  Such  a solid 
would  usually  be  bounded  by  a warped  surface,  but  it  can 
always  be  subdivided  into  one  or  more  of  the  simple  solids 
named  above. 

Inasmuch  as  cylinders  and  cones  are  but  special  forms  of 
prisms  and  pyramids,  and  warped  surface  solids  may  be  divided 
into  elementary  forms  of  them,  and  since  frustums  may  also 
be  subdivided  into  the  elementary  forms,  it  is  sufficient  to  say 
that  all  prismoids  may  be  decomposed  into  prisms,  wedges, 
and  pyramids.  If  a formula  can  be  found  which  is  equally 
applicable  to  all  of  these  forms,  then  it  will  apply  to  any  com- 
bination of  them.  Such  a formula  is  called 

314.  The  Prismoidal  Formula. 

Let  A ==  area  of  the  base  of  a prism,  wedge,  or  pyramid ; 

Afnf  Aj  = the  end  and  middle  areas  of  a prismoid,  or  of  any 
of  its  elementary  solids  ; 

/t  = altitude  of  the  prismoid  or  elementary  solid. 


THE  MEASUREMENT  OF  VOLUMES. 


403 


Then  we  have, 

For  Prisms, 

+ + (I) 


For  Wedges, 


y — 5 (^1  + + ^2) (2) 

For  Pyramids, 

F=y  = §(A  + 4^™  + ^,) (3) 

Whence  for  any  combination  of  these,  having  all  the  common 
altitude  h,  we  have 


F=g(^.+4^„+^,). (4) 

which  is  the  prismoidal  formula. 

It  will  be  noted  that  this  is  a rigid  formula  for  all  prismoids. 
The  only  approximation  involved  in  its  use  is  in  the  assump- 
tion that  the  given  solid  may  be  generated  by  a right  line 
moving  over  the  boundaries  of  the  end  areas. 

This  formula  is  used  for  computing  earthwork  in  cuts  and 
fills  for  railroads,  streets,  highways,  canals,  ditches,  trenches, 
levees,  etc.  In  all  such  cases,  the  shape  of  the  figure  above 
the  natural  surface  in  the  case  of  a fill,  or  below  the  natural 
surface  in  the  case  of  a cut,  is  previously  fixed  upon,  and  to 
complete  the  closed  figure  of  the  several  cross-section  areas 
only  the  outline  of  the  natural  surface  of  the  ground  at  the 
section  remains  to  be  found.  These  sections  should  be  located 
so  near  together  that  the  intervening  solid  may  fairly  be  as- 


404 


SU/^!  VE  Y I NG. 


sumed  to  be  a prisinoid.  They  are  usually  spaced  lOO  feet 
apart,  and  then  intermediate  sections  taken  if  the  irregularities 
seem  to  require  it. 

The  area  of  the  middle  section  is  never  the  mean  of  the 
two  end  areas  if  the  prismoid  contains  any  pyramids  or  cones 
among  its  elementary  forms.  When  the  three  sections  are 
similar  in  form,  the  dimensions  of  the  middle  area  are  always 
the  means  of  the  corresponding  end  dimensions,  d'his  fact 
often  enables  the  dimensions,  and  hence  the  area  of  the  middle 
section,  to  be  computed  from  the  end  areas.  Where  this  can- 
not be  done,  the  middle  section  must  be  measured  on  the 
ground,  or  else  each  alternate  section,  where  they  are  equally 
spaced,  is  taken  as  a middle  section,  and  the  length  of  the 
prismoid  taken  as  twice  the  distance  between  cross-sections. 
For  a continuous  line  of  earthwork,  we  would  then  have,  in 
cubic  yards, 

1+4-^ 2+2^ 3+4^ 4+ 2^ 6+4^ 6 • • +^n)>  • (0 


where  / is  the  distance  between  sections  in  feet.  This  is  the 
same  as  equation  (3),  p.  401.  Here  the  assumption  is.  made 
that  the  volume  lying  between  alternate  sections  conforms 
sufficiently  near  to  the  prismoidal  forms. 

315.  Areas  of  Cross-sections.  — In  most  cases,  in  practice 
at  least,  three  sides  of  a cross-section  are  fixed  by  the  conditions 
of  the  problem.  These  are  the  side  slopes  in  both  cuts  and 
fills,  the  bottom  in  cuts  and  the  top  in  embankments,  or  fills. 
It  then  remains  simply  to  find  where  the  side  slopes  will  cut 
the  natural  surface,  and  also  the  form  of  the  surface  line  on  the 
given  section.  Inasmuch  as  stakes  are  usually  set  at  the  points 
where  the  side  slopes  cut  the  surface,  whether  in  cut  or  fill, 
such  stakes  are  called  slope-stakes,  and  they  are  set  at  the  time 


THE  MEASUREMENT  OF  VOLUMES. 


405 


the  cross-section  is  taken.  The  side  slopes  are  defined  as  so 
much  horizontal  to  one  vertical.  Thus  a slope  of  to  i means 
that  the  horizontal  component  of  a given  portion  of  a slope- 
line is  times  its  vertical  component,  the  horizontal  com- 
ponent always  being  named  first.  The  slope-ratio  is  the  ratio 
of  the  horizontal  to  the  vertical  component,  and  is  therefore 
always  the  same  as  the  first  number  in  the  slope-definition. 
Thus  for  a slope  of  i|  to  i the  slope-ratio  is 

316.  The  Centre  and  Side  Heights. — The  centre  heights 
are  found  from  the  profile  of  the  surface  along  the  centre  line, 
on  which  has  been  drawn  the  grade  line  of  the  proposed  work. 
These  are  carefully  drawn  on  cross-section  paper,  when  the 
height  of  grade  at  each  station  above  or  below  the  surface  line 
can  be  taken  off.  These  centre  heights,  together  with  the 
width  of  base  and  side  slopes  in  cuts  and  in  fills,  are  the  neces- 
sary data  for  fixing  the  position  of  the  slope-stakes.  When 
these  are  set  for  any  section  as  many  points  on  the  surface 
line  joining  them  maybe  taken  as  desired.  In  ordinary  rolling 
ground  usually  no  intermediate  points  are  taken,  the  centre 
point  being  already  determined.  In  this  case  three  points  in 
the  surface  line  are  known,  both  as  to  their  distance  out  from 
the  centre  line  and  as  to  their  height  above  the  grade  line. 
Such  sections  are  called  “ three-level  sections,”  the  surface  lines 
being  assumed  straight  from  the  slope-stakes  to  the  centre 
stake. 

317.  The  Area  of  a Three-level  Section. 

Let  d and  d'  be  the  distances  out,  and 

A and  /i'  the  heights  above  grade  of  right  and  left  slope- 
stakes,  respectively; 

D the  sum  of  d and  d\ 
c the  centre  height, 
r the  slope-ratio, 
w the  width  of  bed. 


4o6 


SUR  VE  YING. 


Then  the  area  ABCDE  is  equal  to  the  sum  of  the  lour  trian- 
gles A£w,  BCw,  wCDy  and  wED.  Or, 


(d  d')  c (Ji  + h'')~ 

^ = i (-) 


This  area  is  also  equal  to  the  sum  of  the  triangles  FCD  and 
FED^  minus  the  triangle  AFB.  Or, 


A = 


id 

4^' 


(2) 


Equation  (2)  can  also  be  obtained  directly  from  equation 

(1)  by  substituting  for  h and  h'  in  (i)  their  values  in  terms  of 

d-^-  ■ 

2 

d and  w,  h — , and  then  putting  D = d-\-  d'.  Equation 

(2)  has  but  two  variables,  c and  D,  and  is  the  most  convenient 
one  to  use. 

318.  Cross-sectioning. — It  will  be  seen  from  Fig.  112  that 
in  the  case  of  a three-level  section  the  only  quantities  to  be 
determined  in  the  field  are  the  heights,  h and  h' , and  the  dis- 
tances out,  d and  d' , of  the  slope-stakes.  These  are  found  by 
trial.  A levelling  instrument  is  set  up  so  as  to  read  on  the 


THE  MEASUREMENT  OF  VOLUMES. 


407 


three  points  C,  D,  E,  and  the  rod  held  first  at  D.  The  reading 
here  gives  the  height  of  instrument  above  this  point.  Add 
this  algebraically  to  the  centre  height  (which  may  be  negative, 
and  which  has  been  obtained  from  the  profile  for  each  station), 
and  the  sum  is  the  height  of  instrument  above  (or  below)  the 
grade  line.  If  the  ground  were  level  transversely,  the  distance 
out  to  the  slope-stakes  would  be 


d=  cr  A — . 
' 2 


But  this  is  not  usually  the  case,  and  hence  the  distance  out 
must  be  found  by  trial.  If  the  ground  slopes  | | 

from  the  centre  line  in  a -j  i the  distance  out  will  evidently 


be  more  than  that  given  by  the  above  equation,  and  vice  versa. 
The  rodman  estimates  this  distance,  and  holds  his  rod  at  a cer- 
tain measured  distance  out,  d^.  The  observer  reads  the  rod, 
and  deducts  the  reading  from  the  height  of  instrument  above 
grade  (or  adds  it  to  the  depth  of  instrument  below  grade),  and 
this  gives  the  height  of  that  point,  above  or  below  grade.  Its 

w 

distance  out,  then,  should  he  d = /qr  + “ • If  this  be  more  than 

the  actual  distance  out,  d^,  the  rod  is  set  farther  in ; if  less,  it  is 
moved  out.  The  whole  operation  is  a very  simple  one  in  prac- 
tice, and  the  rodman  soon  becomes  very  expert  in  estimating 
nearly  the  proper  position  the  first  time. 

In  heavy  work — that  is,  for  large  cuts  or  fills,  and  for  irregu- 
lar ground — it  may  be  necessary  to  take  the  elevation  and  dis- 
tance out  of  other  points  on  the  section  in  order  to  better 
determine  its  area.  These  are  taken  by  simply  reading  on  the 
rod  at  the  critical  points  in  the  outline,  and  measuring  the  dis- 
tances out  from  the  centre.  The  points  can  then  be  plotted 


4o8 


SUKVEVmG. 


on  cross-section  paper  and  joined  by  straight  or  by  free-hand 
curved  lines.  In  the  latter  case  the  area  should  be  deter- 
mined by  planimeter. 

319.  Three-level  Sections,  the  Upper  Surface  con- 
sisting of  two  Warped  Surfaces. — If  the  three  longitudinal 
lines  joining  the  centre  and  side  heights  on  two  adjacent  three- 
level  sections  be  used  as  directrices,  and  two  generatrices,  one 
on  each  side  the  centre,  be  moved  parallel  to  the  end  areas  as 
plane  directers,  two  warped  surfaces  are  generated,  every  cross- 
section  of  which  parallel  to  the  end  areas  is  a three-level  sec- 
tion. These  same  surfaces  could  be  generated  by  two  longi- 
tudinal generatrices,  moving  over  the  surface  end-area  lines  as 
directrices.  In  this  case  the  generating  lines  would  not  move 
parallel  to  a plane  directer,  but  each  would  move  so  as  to  cut 
its  directrices  proportionally.  The  surface  would  therefore  be 
a prismoid,  and  its  exact  volume  would  be  given  by  the  pris- 
moidal  formula.  The  middle  area  in  this  case  is  readily  found, 
since  the  centre  and  side  heights  are  the  means  of  the  corre- 
sponding end  dimensions. 

The  prismoidal  formula. 


could  therefore  be  written 


This  equation  is  derived  directly  from  eq.  (i)  above,  and  eq. 

w 

(2),  p.  406.  The  quantity  — is  the  distance  from  the  grade-plane 


THE  MEASUREMENT  OF  VOLUMES. 


409 


to  the  intersection  of  the  side  slopes,  and  is  a constant  for  any 
given  piece  of  road.  It  would  have  different  values,  however, 
in  cuts  and  fills  on  the  same  line. 

For  brevity,  let 


w 


Here  K is  the  volume  of  the  prism  of  earth,  100  feet  long,  in- 
cluded between  the  roadbed  and  side  slopes.  It  is  first  in- 
cluded in  the  computation  and  then  deducted.  It  is  also  a 
constant  for  a given  piece  of  road. 

Equation  (2)  now  becomes  ' 


where  and  are  the  means  of  and  respectively. 

This  equation  involves  but  two  kinds  of  variables,  c and  D, 
and  is  well  adapted  to  arithmetical,  tabular,  or  graphical  com- 
putation. Thus  if  / =:  100 ; ze/  = 18  ; and  r = i-J ; then  — 
and  K — 200  ; and  equation  (3)  becomes 


^ [(^1  + A + A + ^)A  + 4(^m  + 6)Z>J  — 200  . (4) 


If  the  total  centre  heights  (to  intersection  of  side  slopes)  be 
represented  by  C^,  C„  and  C^,  then  eq.  (3)  becomes,  in  general. 


where  K'  = and  is  independent  of  width  of  bed  and  of 
slopes. 

For  any  given  piece  of  road,  the  constants  K,  K\  and  are 
known,  and  for  each  prismoid  the  Z”s  and  Z^’s  are  observed, 
hence  for  any  prismoid  all  the  quantities  in  eq.  (5)  are  known. 


410 


SUR  VE  YING. 


320.  Construction  of  Tables  for  Prismoidal  Computa- 
tion.— If  a table  were  prepared  giving  the  products  K'CD  for 
various  values  of  C and  it  could  be  used  for  evaluating 
equation  (3),  which  is  the  same  as  equation  (5).  The  argu- 
ments would  be  the  total  widths  (i?,),  and  the  centre  heights 
(6'i).  Such  a table  would  have  to  be  entered  three  times  for 
each  prismoid,  first  with  C,  and  ; second  with  and  ; 
and  finally  with  and  D^.  If  four  times  the  last  tabular 
value  be  added  to  the  sum  of  the  other  two,  and  K subtracted, 
the  result  is  the  true  volume  of  the  prismoid. 


VALUES  OF  (=  AND  K (=  FOR  VARIOUS  WIDTHS 

\ 2;7  \ 4 X 27;/ 

AND  SLOPES. 


Width 

Slopes. 

of 

Road- 

H  to  1. 

to  1. 

to  1. 

1 to  1. 

tol. 

to  1.  194  tol. 

2 to  1. 

bed. 

Q 

AT 

Co 

AT 

Co 

AT 

Co 

AT 

Co 

AT 

Co 

AT 

Co 

A' 

Co 

AT 

10 

20 

370 

10 

185 

6.7 

123 

50 

93 

4.0 

74 

3-3 

62 

2.9 

53 

2-5 

46 

11 

22 

448 

II 

224 

7-3 

149 

5-5 

112 

4.4 

90 

3.7 

75 

3-1 

64 

2.8 

56 

12 

24 

533 

12 

266 

8.0 

178 

6.0 

133 

4.8 

107 

4.0 

89 

3-4 

76 

3.0 

67 

13 

26 

626 

13 

313 

8.7 

209 

6.5 

157 

5-2 

125 

4-3 

104 

3-7 

89 

3-2 

78 

14 

28 

72s 

14 

363 

9-3 

242 

7.0 

181 

5-6 

145 

4.7 

121 

4.0 

104 

3-5 

91 

15 

30 

833 

15 

417 

10. 0 

278 

7-5 

208 

6.0 

167 

5.0 

139 

4-3 

119 

3-8 

104 

16 

32 

948 

16 

474 

10.7 

316 

8.0 

237 

6.4 

190 

5.3 

158 

4.6 

135 

4.0 

118 

17 

34 

1070 

17 

535 

II. 3 

357 

8.5 

268 

6.8 

214 

5-7 

178 

4.9 

153 

4.2 

134 

18 

36 

1200 

18 

600 

12.0 

400 

9.0 

300 

7.2 

240 

6.0 

200 

5-1 

171 

4-5 

150 

19 

38 

1337 

19 

668 

12.7 

446 

9-5 

334 

7.6 

267 

6.3 

223 

4.4 

191 

4.8 

167 

20 

40 

1481 

20 

740 

13.3 

494 

10. 0 

370 

8.0 

296 

6.7 

247 

5-7 

212 

5-0 

185 

21 

42 

1633 

21 

816 

14.0 

544 

10.5 

408 

00 

327 

7.0 

272 

6.0 

233 

5-2 

204 

22 

44 

1793 

22 

896 

14.7 

598 

II  .0 

448 

8.8 

359 

7-3 

299 

6.3 

256 

5-5 

224 

23 

46 

1959 

23 

980 

15-3 

653 

II-5 

490 

9.2 

392 

7-7 

326 

6.6 

280 

5-8 

245 

24 

48 

2134 

24 

1067 

16.0 

711 

12.0 

534 

9.6 

427 

8.0 

356 

6.9 

305 

6.0 

267 

25 

50 

2315 

25 

1158 

16.7 

772 

12. 5 

579 

10. 0 

463 

8.3 

386 

7-1 

331 

6.2 

264 

26 

52 

2504 

26 

1252 

17.3 

835 

13.0 

626 

10.4 

501 

8.7 

417 

7-4 

358 

6.5 

313 

27 

54 

2700 

27 

1350 

18.0 

900 

13.5 

675 

10.8 

540 

9.0 

450 

7-7 

386 

6.8 

338 

28 

56 

2904 

28 

1452 

18.7 

968 

14.0 

726 

II  .2 

581 

9 3 

484 

8.0 

415 

7.0 

363 

29 

58 

3”5 

29 

1558 

19.3 

1038 

14.5 

779 

II. 6 

623 

9.7 

519 

8-3 

445 

7.2 

389 

30 

60 

3333 

30 

1667 

20.0 

nil 

15-0 

833 

12.0 

667 

10. 0 

556 

8.6 

476 

7-5 

417 

THE  MEASUREMENT  OF  VOLUMES. 


All 


Table  XL*  is  such  a table,  computed  for  total  centre  heights 
from  I to  50  feet,  and  for  total  widths  from  i to  100  feet. 
In  railroad  work  neither  of  these  quantities  can  be  as  small  as 
one  foot,  but  the  table  is  designed  for  use  in  all  cases  where 
the  parallel  end  areas  maybe  subdivided  into  an  equal  number 
of  triangles  or  quadrilaterals. 


Example  i.  Three-level  Ground  having  two  Warped  Surfaces. — Find  the 
volume  of  two  prismoids  of  which  the  following  are  the  field-notes,  the  width 
of  bed  being  20  feet,  and  the  slopes  to  i. 


Station  ii. 


Station  12. 


Station  12  -f-  56. 


2S.9f 

0 

43-0 

12.6 

+ 18.6 

-{-  22.0 

27.1 

0 

40.3 

+ 11.4 

20.2 

24-3 

0 

34-9 

+ 9-5 

+ 10.3 

+ 16.6 

From  the  table,  p.  410,  giving  values  of  Co  and  K,  we  find  for  w = 20, 
and  r — i|,  Co  = 6-7,  and  K — 247. 

The  computation  may  be  tabulated  as  follows: 


Sta. 

Width, 

D^d^d'. 

Height, 

C = c + Cq. 

Partial  Volume. 

Volume  of 
Prismoid. 

II 

71.9 

25-3 

562 

M 

69.6 

23-4 

503  X 4 = 2012 

12 

67.4 

21.5 

447 

3021  — 247 

2744 

M 

63-3 

19.2 

374  X 4 = 1496 

12  + 56 

59-2 

17.0 

311 

.56  (2254  - 247) 

1124 

* Modeled  somewhat  after  Crandall’s  Tables,  but  adapted  to  give  volumes 
by  the  Prismoidal  Formula  at  once  instead  of  by  the  method  of  mean  end  areas 
first  and  correcting  by  the  aid  of  another  table  to  give  prismoidal  volumes,  as 
Prof.  Crandall  has  done. 

f The  numerators  are  the  distances  out,  and  the  denominators  are  the  heights 
above  grade,  + denoting  cut  and  — fill. 


412 


SURVEYING. 


Entering^  the  table  (No.  XI.)  fora  width  of  71  and  a height  of  25,  we  find 
548,  to  which  add  7 for  the  3 tenths  of  height,  and  7 more  for  the  9 tenths  in 
width,  both  mentally,  thus  giving  562  cu.  yds.  for  this  partial  volume.  Simi- 
larly for  the  width  67.4,  and  height  21.5,  obtaining  447  cu.  yds.  The  correspond- 
ing result  for  the  middle  area  is  503,  which  is  to  be  multiplied  by  4,  thus  giving 
2012  cu.  yds.  The  sum  of  these  is  3021  cu.  yds.,  from  which  is  to  be  subtracted 
the  constant  volume  A”,  which  in  this  case  is  247  cu.  yds.,  leaving  2774  cu.  yds. 
as  the  volume  of  the  prismoid. 

The  next  prismoid  is  but  56  feet  long,  but  it  is  taken  out  just  the  same  as 
though  it  were  full,  and  then  56  hundredths  of  the  resulting  volume  taken. 
The  data  for  the  12th  station  is  used  in  getting  this  result  without  writing  it 
again  on  the  page. 

Example  2.  Five-level  Ground  having  four  Warped  Surfaces. — Find  the 
volume  of  a prismoid  of  which  the  following  are  the  field-notes,  the  width  of 
bed  being  20  feet,  and  the  slopes  to  i : 


28.9 

150 

0 

20.0 

43-0 

+ 12.6 

+ 12.0 

+ 18.6 

+ 21.0 

+ 22.0 

27.1 

12.5 

0 

18.5 

40.3 

+ 11. 4 

+ 12.0 

+ 14.8 

+ 19.6 

+20.2 

This  is  the  same  problem  as  the  preceding,  with  intermediate  heights 
added. 

To  compute  this  from  the  table,  it  is  separated  into  three  prismoids,  as  shown 
in  Fig.  1 13. 


Fig.  113. 


Let  ABDGCFE  be  the  cross-section.  This  may  be  separated  into  the  triangle 
ABC,  and  the  two  quadrilaterals  BCGD  and  ACFE.  The  area  of  the  triangle  is 
icw.  That  of  the  right  quadrilateral  is,  from  Art.  179,  p.  202, 


THE  MEASUREMENT  OF  VOLUMES. 


\\2a 


c (^dic  — — o)  + — dky^  = — A)(^dk  — + kdn  ^ 


Similarly  the  area  of  the  left  quadrilateral  is  ^ 
The  total  area  of  the  section  then  is 


+ lid'n  CW  kdn  + {f 


. . (I) 


If  the  interior  side  elevations  be  taken  over  the  edges  of  the  base,  then 

*W  'W 

d'k and  dk both  become  zero,  and  the  first  and  last  terms  disappear. 

2 2 

Or  if  the  centre  and  extreme  side  heights  are  the  same,  these  terms  go  out. 
Experience  shows  that  these  terms  can  usually  be  neglected  without  material 
error.  If  they  are  retained,  each  partial  volume  will  be  composed  of  five  terms, 
while  if  they  are  neglected  there  will  be  but  three.  The  signs  of  these  terms  also 
must  be  carefully  attended  to.  When  the  interior  side  readings  are  taken  over  the 
edges  of  the  base,  therefore,  this  equation  becomes 


A = i {k'd'h  + CW  + kdh) 


(2) 


The  tables  are  well  adapted  to  compute  the  prismoidal  volume  for  five-level 
sections  by  either  of  these  formulae.  Thus,  if  the  adjacent  section  also  has  five 
points  determined  in  its  surface,  its  area  may  be  represented  by  an  equation  similar 
to  one  of  these,  and  from  these  end-area  data  mean  values  may  be  found  for  the 
corresponding  middle-area  points,  and  the  volumes  taken  out  as  before.  In  this 
case  the  prism  included  between  the  road-bed  and  side- slopes,  whose  volume  is  K, 
is  not  included,  and  hence  its  volume  is  not  to  be  deducted  from  the  result.  The 
computation  by  table  XI.  of  equation  (i)  would  be  as  follows  : 


Sta. 

k'. 

d'k. 

k'. 

d'u- 

c. 

4. 

k. 

dh- 

h. 

Partial  Volumes. 

Total 

Volume. 

II 

12.6 

28.9 

12.0 

15.0 

18.6 

20.0 

21.0 

43-0 

22,0 

■f9  + 108 -(- 114 -1-279— 10  = 500 

M 

12.0 

28.0 

12.0 

13.8 

16.7 

19.2 

20.3 

41 .6 

21. I 

4( -J- 6 -f- 104  + 102 -t- 260  - 1 2) = 1 840 

12 

II. 4 

1 

27.1 

1 

12.0 

12.5 

14.8 

18. s 

19.6 

40-3 

20.2 

-1-3  + 100+  90  + 242—13=  422 

2762 

412^ 


SUR  VE  YING. 


The  use  of  the  table  is  the  same  as  before.  First  take  out  from  the  table  the 


volume  corresponding  to  (c  ~ Ji) 


which  when  evaluated  for  section  ii 


is  (i8.6  — i2.6)(i5.o—  io)  = 6.0  x 5.0.  This  is  positive,  and  the  volume  corre- 
sponding to  a depth  of  6.0  feet  and  a width  of  5.0  feet  is  9 cubic  yards.  Proceed 
to  evaluate  the  remaining  terms  of  eq.  (i)  in  a similar  manner,  the  last  term 
coming  out  negative.  The  dimensions  of  the  mid  section  are  the  means  of  the 
corresponding  end  dimensions,  as  before.  If  one  end-area  is  a three-level  section 
and  the  next  a five-level  section,  the  included  prismoid  is  computed  as  a five-level 
prismoid,  the  vanishing  points  in  the  three-level  section  corresponding  to  the 
interior  side  elevations  on  the  five-level  section  being  indicated  in  the  field.  Par- 
tial stations,  or  prismoids,  are  first  computed  as  though  they  were  100  feet  long 
(for  which  the  table  is  constructed),  and  then  multiplied  by  their  length  and  divided 
by  100  as  before. 

If  equation  (2)  may  be  used,  the  work  is  shortened  very  much.  The  columns 
in  //',  d\  , dk  , and  /^,  may  be  omitted,  and  there  will  also  be  but  three  terms  in 
each  partial  product.  Thus,  if  sections  ii  and  12  had  been  taken  with  the  interior 
elevations,  each  10  feet  from  the  centre  line,  we  might  have  had  something  as 
follows  : 


28.9 

lO.O 

0 

10. 0 

43-0 

-f-  12.6 

+ 15-4 

-f  i8.6 

-b  19.8 

-f-  22.0 

27»i 

10. 0 

0 

10. 0 

40.3 

-f-11.4 

+ 12.5 

+ 14.8 

+ 17-4 

-1-20.2 

The  computation  then,  by  eq.  (2),  would  have  been  : 


Sta. 

d\. 

k'. 

c. 

k. 

dh- 

Partial  Volumes. 

Total 

Volume. 

II 

28.9 

15-4 

18.6 

19.8 

43-0 

137  + II4  + 263  = 514 

M 

28.0 

14.0 

16.7 

18.6 

41.6 

4 (121  + 102  -f  239)  = 1848 

12 

27.1 

12.5 

14.8 

17.4 

40.3 

104  -f  90  -f  215  = 409 

2771 

By  this  method  the  computation  of  a five-level  section  is  little  more  trouble 


THE  MEASUREMENT  OF  VOLUMES. 


413 


than  that  of  a three-level  section,  and  yet  the  intermediate  points  taken  at  a dis- 
w 

tance  of  — from  the  centre,  are  apt  to  increase  the  accuracy  considerably  on 
ordinary  rolling  ground. 


321.  Three-level  Sections,  the  Surface  divided  into 
four  Planes  by  Diagonals. — If  the  surface  included  between 
two  three-level  sections  be  assumed  to  be  made  up  of  four 
planes  formed  by  joining  the  centre  height  at  one  end  with  a 
side  height  at  the  other  end  sec- 
tion on  each  side  the  centre  line 
(Fig.  1 14),  these  lines  being  called 
diagonals,  an  exact  computation  of 
the  volume  is  readily  made  without 
computing  the  mid-area.  Two  diag- 
onals are  possible  on  each  side  the 
centre  line  but  the  one  is  drawn 
which  is  observed  to  most  nearly 
fit  the  surface.  They  are  noted  in 
the  field  when  the  cross-sections  are 
taken. 

The  total  volume  of  such  a prismoid  in  cubic  * yards  is 
V =■  ^ 2y  ^1  Vl  T (^2  d"  ^2  )^2  d“  T D’ C* 

+ + + V + + (I) 

where  and  h-l  are  the  centre  and  side  heights  at  one  sec- 
tion and  dx  and  d(  the  distances  out,  d^,  and  d<^  be- 


* For  a demonstration  of  this  formula  see  Henck’s  Field-Book. 


414 


SUR  VE  YING. 


ing  the  corresponding  values  for  the  other  end  section.  C and 
C are  the  centre  heights,  //  and  //'  the  side  heights,  and  D 
and  D'  the  distances  out  on  the  right  and  left  diagonals. 
Although  this  formula  seems  long,  the  computations  by  it  arc 
very  simple.  Thus  let  the  volume  be  found  from  the  following 
field-notes  for  a base  of  20  feet  and  side  slopes  to  i. 


22 

0 

47*5 

+ 8\ 

4-8\ 

+ 25* 

34 

\ ° 

\ 

+ 16 

+ 4 

+ 4* 

The  upper  figures  indicate  the  distances  out  and  those 
below  the  lines  the  heights,  the  plus  sign  being  used  for  cuts. 
The  computation  in  tabular  form  is  as  follows : 


Sta. 

d. 

h. 

c. 

h'. 

d'. 

d-\rd'. 

{d^d')c. 

DC. 

D’C. 

I 

22 

8 

8 

25 

47-5 

69.5 

556 

.... 

• • « • 

2 

34 

16 

4 

4 

16 

50.0 

200 

88 

128 

h\  -|-  h-i  — 24  88 

= 12  128 

s = 65  X 10  = 650 


6 ) 162200 
27  ) 27033 

1001  cu.  yards. 

The  great  advantage  of  the  method  consists  in  the  data> 
all  being  at  hand  in  the  field-notes. 

Hudson’s  Tables*  give  volumes  for  this  kind  of  prismoid. 

* Tables  for  Computing  the  Cubic  Contents  of  Excavations  and  Embank- 
ments. By  John  R.  Hudson,  C.E.  John  Wiley  & Sons,  New  York,  1884. 


THE  MEASUREMENT  OF  VOLUMES. 


415 


They  furnish  a very  ready  method  of  computing  volumes  when 
this  system  is  used, 

322.  Comparison  of  Methods  by  Diagonals  and  by 
Warped  Surfaces. — Although  the  surveyor  has  a choice  of 
two  sets  of  diagonals  when  this  method  is  used,  the  real  surface 
would  usually  correspond  much  nearer  the  mean  of  the  two  pairs 
of  plane  surfaces  than  to  either  one  of  them.  That  is,  the 
natural  surface  is  curved  and  not  angular,  and  therefore  it  is 
probable  that  two  warped  surfaces  joining  two  three-level  sec- 
tions would  generally  fit  the  ground  better  than  four  planes, 
notwithstanding  the  choice  that  is  allowed  in  the  fitting  of  the 
planes.  More  especially  must  this  be  granted  when  the  truth 
of  the  following  proposition  is  established. 

Proposition  : The  volume  mdiidcd  between  two  three-level 
sections  having  their  corresponding  surface  lines  joined  by 
warped  surfaces^  is  exactly  a mean  betweeji  the  two  volumes 
formed  between  the  same  end  sections  by  the  two  sets  of  planes  re- 
sulting from  the  two  sets  of  diagonals  which  jnay  be  drawn. 

If  the  two  sets  of  diagonals  be  drawn  on  each  side  the 
centre  line  and  a cross-section  be  taken  parallel  to  the  end 
areas,  the  traces  of  the  four  surface  planes  on  each  side  the 
centre  line  on  the  cutting  plane  will  form  a parallelogram, 
the  diagonal  of  which  is  the  trace  of  the  warped  surface  on 
this  cutting  plane.  Since  this  cutting  plane  is  any  plane  par- 
allel to  the  end  areas,  and  since  the  warped  surface  line  bisects 
the  figure  formed  by  the  two  sets  of  planes  formed  by  the 
diagonals,  it  follows  that  the  warped  surface  bisects  the  volume 
formed  by  the  two  sets  of  planes.  The  proposition  will  there- 
fore be  established  if  it  be  shown  that  the  trace  of  the  warped 
surface  is  the  diagonal  of  the  parallelogram  formed  by  the 
traces  of  the  four  planes  formed  by  the  two  sets  of  diagonals. 
Fig.  1 15  shows  an  extreme  case  where  the  centre  height  is 
higher  than  the  side  height  at  one  end  and  lower  at  the  other. 
Only  the  left  half  of  the  prismoid  is  shown  in  the  figure.  The 


4i6 


SUJ?  VE  Y TNG. 


cutting  plane  cuts  the  centre  and  side  lines  and  the  two  diago- 
nals in  cfgh  on  the  plane,  and  in  c'fg'h'  on  the  vertical 
projection.  For  the  diagonal  the  surface  lines  cut  out  are 

e'f'  and  f'h'.  For  the  diagonal  they  are  e'g'  and  g'h\ 
For  the  warped  surface  the  line  cut  out  is  e'h\  this  being  an 


Fig.  115. 

element  of  that  surface.  It  remains  to  show  that  e'fdi'g'  is  a 
parallelogram. 

Since  the  cutting  plane  is  parallel  to  the  end  planes  all  the 
lines  cut  are  divided  proportionally.  That  is,  if  the  cutting 
plane  is  one  of  I from  then  it  cuts  off  one  of  all  the 
lines  cut,  measured  from  that  end  plane.  But  if  the  lines 
are  divided  proportionally,  the  projections  of  those  lines  are 
divided  proportionally,  and  hence  the  points  e' ,f  divide 


THE  MEASUREMENT  OF  VOLUMES. 


417 


the  sides  of  the  quadrilateral  proportionally.  But 

it  is  a proposition  in  geometry  that  if  the  four  sides  of  a quad- 
rilateral, or  two  opposite  sides  and  the  diagonals,  be  divided 
proportionally  and  the  corresponding  points  of  subdivision 
joined,  the  resulting  figure  is  a parallelogram.  Therefore  ef'H 
g'  is  a parallelogram,  and  e'li  is  one  of  its  diagonals  and  hence 
bisects  it.  Whence  the  surface  generated  by  this  line  moving 
along  and  parallel  to  the  end  areas  bisects  the  volume 
formed  by  the  four  planes  resulting  from  the  use  of  both  di- 
agonals on  one  side  the  centre  line.  Q.  E.  D. 

It  is  probable,  therefore,  that  the  warped  surface  would 
usually  fit  the  ground  better  than  either  of  the  sets  of  planes 
formed  by  the  diagonals.  Furthermore,  the  errors  caused  by 
the  use  of  the  warped  surface  (Table  XL)  are  compensating 
errors,  thus  preventing  any  marked  accumulation  of  errors  in 
a series  of  prismoids.*  There  are  extreme  cases,  however, 
such  as  that  given  in  the  example,  Fig.  114,  which  are  best 
computed  by  the  method  by  diagonals. 

323.  Preliminary  Estimate  from  the  Profile. — If  the 
cross-sections  be  assumed  level  transversely  then  for  given 
width  of  bed  and  side  slopes,  a table  of  end  areas  may  be  pre- 
pared in  terms  of  the  centre  heights.  From  such  a table  the 


* The  two  methods  here  discussed  are  the  only  ones  that  have  any  claims  to 
accuracy.  The  method  by  “ mean  end  areas,”  wherein  the  volume  is  assumed 
to  be  the  mean  of  the  end  areas  into  the  length,  always  gives  too  great  a volume 
(except  when  a greater  centre  height  is  found  in  connection  with  a less  total 
width,  which  seldom  occurs),  the  excess  being  one  sixth  of  the  volume  of  the 
pyramids  involved  in  the  elementary  forms  of  the  prismoid.  This  is  a large  error 
even  in  level  sections,  and  very  much  greater  on  sloping  ground,  and  yet 
it  is  the  basis  of  most  of  the  tables  used  in  computing  earthwork,  and  in  some 
States  it  is  legalized  by  statute.  Thus  in  the  example  computed  by  Henck’s 
method  on  p.  414  the  volume  by  mean  end  areas  is  1193  cu.  yards;  by  the 
prismoidal  formula  u is  1168  cu.  yards,  while  by  the  method  by  diagonals  it  was 
only  1001  cu.  yards.  This  was  an  extreme  case,  however,  and  was  selected  to 
show  the  adaptation  of  the  method  by  diagonals  to  such  a form. 

27 


4i8 


SURVEYING. 


end  areas  may  be  rapidly  taken  out  and  plotted  as  ordinates 


from  the  grade  line.  The  ends  of  these  ordinates  may  then 
be  joined  by  a free-hand  curve,  and  the  area  of  this  curve 
found  by  the  planimeter.  The  ordinates  may  be  plotted  to 
such  a scale  that  each  unit  of  the  area,  as  one  square  inch, 
shall  represent  a convenient  number  of  cubic  yards,  as  looo. 
The  record  of  the  planimeter  then  in  square  inches  and  thou- 
sandths gives  at  once  the  cubic  yards  on  the  entire  length  of 
line  worked  over  by  simply  omitting  the  decimal  point.  Evi- 
dently the  scale  to  which  the  ordinates  are  to  be  drawn  to  give 
such  a result  is  not  only  a function  of  the  width  of  bed  and 
side  slopes,  but  also  of  the  longitudinal  scale  to  which  the  pro- 
file line  is  plotted.  The  area  of  a level  section  is 


A = we rc^,  . . 


• • (I) 


where  w.,  and  r are  the  width  of  base,  centre  height,  and 
slope-ratio  respectively. 

Now  if  li  — the  horizontal  scale  of  the  profile,  that  is  the 
number  of  feet  to  the  inch,  and  if  one  square  inch  of  area  is  to 
represent  lOOO  cu.  yards,  the  length  of  the  ordinate  must  be 


hA  h {zve  -|-  rc"^) 


(2) 


lOOO  X 27  27,CXX) 


If  values  be  given  to  //,  and  r,  which  are  constants  for 
any  given  case,  then  the  value  of  y becomes  a function  of  c 
only,  and  a table  can  be  easily  prepared  for  the  case  in  hand. 
Since  y is  a.  function  of  the  second  power  of  e,  the  second  dif- 
ference will  be  a constant,  and  the  table  can  be  prepared  by 
means  of  first  and  second  differences.  Thus  if  c takes  a small 
increment,  as  i foot,  then  the  first  difference  is 


(3) 


THE  MEASUREMENT  OF  VOLUMES. 


419 


But  this  first  difference  is  also  a function  of  c,  and  hence  when 
c takes  an  increment  this  first  difference  changes  by  an  amount 
equal  to 


A'y  = 


h 

27000 


2r, 


(4) 


which  is  constant.  An  initial  first  difference  being  given  for  a 
certain  value  of  a column  of  first  differences  can  be  obtained 
by  simply  adding  the  A"y  continuously  to  the  preceding  sum. 
With  this  column  of  first  differences  the  corresponding  column 
of  values  of  y may  be  found  by  adding  the  first  differences  con- 
tinuously to  the  initial  value  of  y for  that  column.* 


TABULAR  VALUES  OF  jy  IN  EQUATION  (2)  FOR  w=2o,  r=ii,  AND 

h — 400. 


c 

o.'o 

O.'l 

0.'2 

o.'3 

o.'4 

o.'5 

o.'6 

o.'l 

o.'8 

o.'9 

in. 

in. 

in. 

in. 

in. 

in. 

in. 

in. 

in. 

in. 

0 

0.00 

0.03 

0.06 

0.09 

0.12 

0.15 

0. 19 

0.22 

0.25 

0.28 

I 

•32 

•35 

-39 

.42 

.46 

-49 

-.53 

-57 

.61 

.64 

2 

.68 

.72 

-76 

.80 

.84 

.88 

.92 

.96 

1. 00 

1.05 

3 

1.09 

1-13 

1-17 

1.22 

1.26 

1-31 

1-35 

1.40 

1-45 

1-49 

4 

1-54 

1.59 

1.63 

1 .69 

1-73 

1.78 

1.83 

1.88 

1.93 

1-99 

5 

2.04 

2.09 

2.14 

2.19 

2.24 

2.30 

2.36 

2.41 

2.47 

3-52 

6 

2.58 

2.63 

2.69 

2.75 

2.80 

2.87 

2.92 

2.98 

3-04 

3.10 

7 

316 

3.22 

3.28 

3-35 

3-41 

3-47 

3-54 

3.60 

3.66 

3-73 

8 

3-79 

3.86 

3-92 

3-99 

405 

4-13 

4-19 

4.26 

4.33 

4-40 

9 

4-47 

4-54 

4.60 

4.68 

4-75 

4.82 

4-89 

4-97 

5-04 

5.11 

10 

5.18 

5.26 

5-33 

5-40 

5 48 

5-56 

5-64 

5-72 

5-79 

5-87 

II 

5-95 

6.03 

6.10 

6.18 

6.26 

6.35 

6-43 

6.51 

6.59 

6.67 

12 

6.76 

6.84 

6.92 

7.00 

7-09 

7.18 

7.26 

7.35 

7-43 

7-52 

13 

7.61 

7.70 

7.78 

7.86 

7.96 

8.05 

8.14 

8.23 

8.32 

8.41 

14 

8.50 

8.60 

8.68 

8.77 

8.87 

8.97 

9.06 

9.16 

9-25 

9-35 

15 

9.44 

9-54 

9-63 

9-73 

9-83 

9-94 

10.03 

10.13 

10.23 

10.33 

16 

10.43 

10-53 

10.62 

10.73 

10.83 

10.94 

11.04 

11.15 

11.25 

11-35 

17 

11.46 

11.56 

11.66 

11-77 

11.88 

12.00 

12.10 

12.21 

12.31 

12.42 

18 

12.53 

12.64 

12.75 

12.86 

12.97 

13-09 

13.20 

13.32 

13.42 

13-54 

19 

13-65 

13-77 

13.87 

13-99 

14.10 

14  23 

14.34 

14-47 

14.58 

14-70 

20 

14.81 

14.93 

15-04 

15.16 

15.29 

15-42 

15.53 

15.66 

15.78 

15.90 

* For  a further  exposition  of  this  subject,  see  Appendix  C. 


420 


SURVEYING. 


The  preceding  table  was  constructed  in  this  manner,  for 
w — 20  feet,  r = \\\  and  h — 400  feet  to  the  inch. 

324.  Borrow-pits  are  excavations  from  which  earth  has 
been  “ borrowed  ” to  make  an  embankment.  It  is  generally 
preferable  to  measure  the  earth  in  cut  rather  than  in  fill,  hence 
when  the  earth  is  taken  from  borrow-pits  and  its  volume  is  to 
be  computed  in  cut,  the  pits  must  be  carefully  staked  out  and 
elevations  taken  both  before  and  after  excavating.  The  meth- 
ods given  in  art.  31 1 are  well  suited  to  this  purpose,  or  they 
may  be  computed  as  prismoids  by  the  aid  of  Table  XL,  if  pre- 
ferred. To  use  the  table  it  is  only  necessary  to  enter  it  with 
such  heights  and  widths  as  give  twice  the  elementary  areas 
(triangles  or  quadrilaterals)  into  which  the  end  sections  are 
divided,  and  then  multiply  the  final  result  by  the  length  and 
divide  by  100.  The  table  is  entered  for  both  end-area  dimen- 
sions and  also  the  mid-area  dimensions,  four  times  this  latter 
result  being  taken  the  same  as  before. 

325.  Shrinkage  of  Earthvvork. — Excavated  earth  first 
increases  in  volume,  when  removed  from  a cut  and  dumped  on 
a fill,  but  it  gradually  settles,  or  shrinks,  until  it  finally  comes 
to  occupy  a less  volume  than  it  formerly  did  in  the  cut.  Both 
the  amounts,  initial  increase,  and  final  shrinkage  depend  on  the 
nature  of  the  soil,  its  condition  when  removed,  and  the  man- 
ner of  depositing  it  in  place.  There  can  therefore  be  no  gen- 
eral rules  given  which  will  always  apply.  For  ordinary  clay 
and  sandy  loam^  dumped  loosely,  the  first  increase  is  about  one 
twelfth,  and  then  the  settlement  about  one  sixth  of  this  increased 
volume,  leaving  a final  volume  of  about  nme  tenths  of  the  original 
volume  in  cut.^ 

Thus  for  100  cubic  yards  of  settled  embankment  in  cubic 
yards  in  cut  would  be  required.  But  a contractor  should  have 


* See  paper  by  P.  J.  Flynn  in  Trans.  Tech.  Soc.  of  the  Pacific  Coast,  vol. 
ii.  p.  179,  where  all  the  available  experimental  data  are  given. 


THE  MEASUREMENT  OF  VOLUMES. 


421 


his  stakes  or  poles  set  one  fifth  higher  than  the  corresponding 
fill,  so  that  when  filled  to  the  tops  of  these,  a settlement  of 
one  sixth  will  bring  the  surface  to  the  required  grade. 

These  changes  of  volume  are  less  for  sand  and  more  for 
stiff,  wet  clay. 

For  rock  the  permanent  increase  in  volume  is  from  60  to 
80  per  cent,  the  greater  increase  corresponding  to  a smaller 
average  size  of  fragment. 

326.  Excavations  under  Water. — It  is  often  necessary  to 
determine  the  volume  of  earth,  sand,  mud,  or  rock  removed 
from  the  beds  of  rivers,  harbors,  canals,  etc.  If  this  be  done 
by  soundings  alone,  it  is  likely  to  work  injustice  to  the  con- 
tractor, as  he  would  receive  no  pay  for  depths  excavated  below 
the  required  limit ; and  besides,  foreign  material  is  apt  to  flow 
in  and  partially  replace  what  is  removed,  so  that  the  material 
actually  excavated  is  not  adequately  shown  by  soundings 
within  the  required  limits.  It  is  common,  therefore,  to  pay 
for  the  material  actually  removed,  an  inspector  being  usually 
furnished  by  the  employer  to  see  that  no  useless  work  is  done 
beyond  the  proper  bounds.  The  material  is  then  measured  in 
the  dumping  scows  or  barges.  The  unit  of  measure  is  the 
cubic  yard,  the  same  as  in  earthwork.  There  are  two  general 
methods  of  gauging  scows,  or  boats.  One  is  to  actually  meas- 
ure the  inside  dimensions  of  each  load,  which  is  often  done  in 
the  case  of  rock,  and  the  other  is  to  measure  the  displacement 
of  the  boat,  which  is  the  more  common  method  with  dredged 
material.  When  the  barge  is  gauged  by  measuring  its  dis- 
placement, the  water  in  the  hold  must  always  be  pumped  down 
to  a given  level,  or  else  it  must  be  gauged  both  before  and  after 
loading  and  the  depth  of  water  in  the  hold  observed  at  each 
gauging.  A displacement  diagram  (or  table)  is  prepared  for 
each  barge,  from  its  actual  external  dimensions,  in  terms  of  its 
mean  draught.  There  should  always  be  four  gaugings  taken 
to  determine  the  draught,  at  four  symmetrically  located  points 


422 


SURVEYING. 


on  the  sides,  these  being  one  fourth  the  length  of  the  barge 
from  the  ends.  Fixed  gauge-scales,  reading  to  feet  and  tenths 
may  be  painted  on  the  side  of  the  barge,  or  if  it  is  flat-bot- 
tomed, a gauging-rod,  with  a hook  on  its  lower  end  at  the  zero 
of  the  scale,  may  be  used  and  readings  taken  at  these  four 
points.  Any  distortion  of  the  barge  under  its  load,  or  any 
unsymmetrical  loading,  will  then  be  allowed  for,  the  mean  of 
the  four  gauge-readings  being  the  true  mean  draught  of  the 
boat. 

To  prepare  a displacement  diagram,  the  areas  of  the  sur- 
faces of  displacement  must  be  found  for  a series  of  depths  uni- 
formly spaced.  This  series  may  begin  with  the  depth  for  no 
load,  the  hold  being  dry.  They  should  then  be  found  for  each 
five  tenths  of  a foot  up  to  the  maximum  draught.  If  the  boat 
has  plane  vertical  sides  and  sloped  ends  these  areas  are  rec- 
tangles, and  are  readily  computed.  If  the  boat  is  modelled  to 
curved  lines,  the  water-lines  can  be  obtained  from  the  original 
drawings  of  the  boat,  or  else  they  must  be  obtained  by  actual 
measurement.  In  either  case  they  can  be  plotted  on  paper, 
and  their  areas  determined  by  a planimeter.  These  areas  are 
analogous  to  the  cross-sections  in  the  case  of  railroad  earth- 
work, and  the  prismoidal  formula  may  be  applied  for  comput- 
ing the  displacement.  Thus, 

Let  A^,  A^,  etc.,  be  the  areas  of  the  displaced  water 

surfaces,  taken  at  uniform  vertical  distances  h apart.  Then 
for  an  even  number  of  intervals  we  have  in  cubic  yards 

F=  + . (I) 

If  the  total  range  in  draught  be  divided  into  six  equal  por- 
tions, each  equal  to  h,  then  Weddel’s  Rule  * would  give  a 


For  the  derivation  of  this  rule  see  Appendix  C. 


THE  MEASUREMENT  OF  VOLUMES 


423 


nearer  approximation.  With  the  same  notation  as  the  above 
we  would  then  have,  in  cubic  yards, 

= ~ [^0  + + ^4  + ^6  + 5 (^i  + ^3  + ^5)  + ^3]-  • (2) 

These  rules  are  also  applicable  to  the  gauging  of  reservoirs, 
mill-ponds,  or  of  any  irregular  volume  or  cavity. 

After  the  displaced  volume  of  water  is  found,  the  corre- 
sponding volume  of  earth  or  rock  is  found  by  applying  a proper 
constant  coefficient.  This  coefficient  is  always  less  than  unity, 
and  is  the  reciprocal  of  the  specific  gravity  of  the  material. 
This  must  be  found  by  experiment.  In  the  case  of  soft  mud 
it  is  nearly  unity,  while  with  sand  and  rock  it  is  much  more. 
When  rock  is  purchased  by  the  cubic  yard,  solid  rock  is  not 
implied,  but  the  given  quality  of  cut  or  roughly-quarried  rock, 
piled  as  closely  as  possible.  When  rock  is  excavated,  solid 
rock  is  meant.  A measured  volume  of  any  material  put  into  a 
gauged  scow  will  give  the  proper  coefficient  for  that  material. 
Thus  if  the  measured  volume  V give  a displacement  of  F, 
V 

then  — C \s  the  coefficient  to  apply  to  the  displacement  to 
give  the  volume  of  that  material. 


CHAPTER  XIV. 

GEODETIC  SURVEYING. 

327.  The  Objects  of  a Geodetic  Survey  are  to  accurately 
determine  the  relative  positions  of  widely  separated  points  on 
the  earth’s  surface  and  the  directions  and  lengths  of  the  lines 
joining  them  ; or  to  accurately  determine  the  absolute 
(in  latitude,  in  longitude  from  a fixed  meridian,  and  in  eleva- 
tion above  the  sea-level)  of  widely  separated  points  on  the 
earth’s  surface  and  the  directions  and  lengths  of  the  lines  join- 
ing them. 

In  the  first  case  the  work  serves  simply  t©  supply  a skeleton 
of  exact  distances  and  directions  on  which  to  base  a more  de- 
tailed survey  of  the  intervening  country  ; in  the  second,  the  re- 
sults furnish  the  data  for  computing  the  shape  and  size  of  the 
earth,  in  addition  to  their  use  in  more  detailed  surveys. 

It  is  usually  desirable  also  to  have  some  knowledge  of  the 
latitude  and  longitude  of  the  points  determined  in  the  first 
case,  but  a very  accurate  knowledge  of  these  would  not  be  es- 
sential to  the  immediate  objects  of  the  work. 

In  both  cases  the  points  determined  form  the  vertices  of  a 
series  of  triangles  joining  all  the  points  in  the  system.  One  or 
more  lines  in  this  system  of  triangles  and  all  of  the  angles  are 
very  carefully  measured,  and  the  lengths  of  all  other  lines  in 
the  system  computed.  The  azimuths  of  certain  lines  are  also 
determined,  and,  if  desired,  the  latitudes  and  longitudes  of  some 
of  the  points.  I"rom  this  data  it  is  then  possible  to  compute 
the  latitudes  and  longitudes  of  all  the  points  in  the  system  and 


GEODETIC  SURVEYING. 


425 


the  lengths  and  azimuths  of  all  the  connecting  lines.  The 
work  as  a whole  is  denominated  triangulation. 

The  measured  lines  are  called  base-lines,  the  points  deter- 
mined are  triangulation-stations,  and  those  points  (usually  tri- 
angulation-stations) at  which  latitude,  longitude,  or  azimuth 
is  directly  determined  are  called  respectively  latitude,  longi- 
tude, or  azimuth  stations.  The  latitude  of  a station  and  the 
azimuth  of  a line  are  determined  at  once  by  stellar  observations 
at  the  point.  The  longitude  is  found  by  observing  the  differ- 
ence of  time  elapsing  between  the  transit  of  a star  across  the 
meridian  of  the  longitude-station  and  the  meridian  of  some 
fixed  observatory  whose  longitude  is  well  determined.  An  ob- 
server at  each  station  notes  the  time  of  transit  across  his  merid- 
ian, and  each  transit  is  recorded  upon  a chronograph-sheet  at 
each  station.  This  requires  a continuous  electrical  connection 
between  the  two  stations.  This  difference  of  time,  changed 
into  longitude,  gives  the  longitude  of  the  field-station  with  ref- 
erence to  the  observatory. 

328.  Triangulation  Systems  are  of  all  degrees  of  magni- 
tude and  accuracy,  from  the  single  triangle  introduced  into  a 
course  to  pass  an  obstruction,  up  to  the  large  primary  systems 
covering  entire  continents,  the  single  lines  in  which  are  some- 
times over  one  hundred  miles  in  length. 

The  methods  herein  described  will  apply  especially  to  what 
might  be  called  secondary  and  tertiary  systems,  the  lines  of 
which  are  from  one  to  twenty  miles  in  length,  and  the  accu- 
racy of  the  work  anywhere  from  i in  5000  to  i in  50,000.  Al- 
though the  methods  used  are  more  or  less  common  to  all  sys- 
tems, yet  for  the  primary  systems,  where  great  areas  are  to  be 
covered  and  the  highest  attainable  accuracy  secured,  many 
refinements,  both  in  field  methods  and  in  the  reductions,  are 
introduced  which  would  be  found  useless  or  needlessly  expen- 
sive in  smaller  systems. 

If  it  is  desired  to  connect  two  distant  points  by  a system 


426 


SURVEYING. 


of  triangulation  at  the  least  expense,  then  use  system  I.,sliown 
in  Fig.  1 16.  This  S3^stcm  is  also  adapted  to  the  fixing  of  a 
double  row  of  stations  with  the  least  labor. 

If  such  distant  points  are  to  be  joined,  or  such  double  system 
of  stations  established,  with  the  greatest  attainable  accuracy, 
then  system  III.  should  be  used.  This  system  is  also  best 
adapted  to  secondary  work,  where  it  is  desired  to  simplify  the 
work  of  reduction.  Each  quadrilateral  is  independently  re- 
duced. 

If  the  greatest  area  is  to  be  covered  for  a given  degree  of 
accuracy  or  cost,  then  system  II.  is  the  one  to  use. 

System  I.  consists  of  a single  row  of  simple  triangles,  sys- 


I. 


tern  II.  of  a double  row  of  simple  triangles  or  of  simple  tri- 
angles arranged  as  hexagons,  and  system  III.  of  a single  row 
of  quadrilaterals.  A quadrilateral  in  triangulation  is  an  arrange- 
ment of  four  stations  with  all  the  connecting  lines  observed. 
This  gives  six  lines  connecting  as  many  pairs  of  stations,  over 
which  pointings  have  been  taken  from  both  ends  of  the  line. 


GEODETIC  SURVEYING. 


42^ 


For  the  same  maximum  length  of  lines  we  have  the  follow- 
ing comparison  of  the  three  systems : * 


System. 

Composition. 

Distance 

Covered. 

No.  of 
Sta- 
tions. 

Total 

Length 

of 

Sides. 

Area 

Covered. 

No.  of 
Conditions. 

I. 

Equilateral  triangles. 

5 

II 

19 

4.5 

O' 

II 

1 

II. 

Hexagons. 

5.2 

17 

34 

9 

5 

III. 

Quadrilaterals  (squares). 

4.95 

16 

29 

3-5 

2n  — 4 = 28 

Thus,  for  the  same  distance  covered,  the  number  of  sta- 
tions to  be  occupied  and  the  total  length  of  lines  to  be  cleared 
out  are  about  one  half  more  for  systems  II.  and  III.  than  for 
system  I.  The  area  covered  by  system  II.  is  twice  that  by 
system  I.,  but  the  number  of  conditions  is  much  greater  in 
system  III.  than  in  either  of  the  others.  Since  almost  all  the 
error  in  triangulation  comes  from  erroneous  angle-measure- 
ments, the  results  will  be  more  accurate  according  to  our 
ability  to  reduce  the  observed  values  of  the  angles  to  their 
true  values.  The  “ conditions”  mentioned  in  the  above  table 
are  rigid  geometrical  conditions,  which  must  be  fulfilled  (as 
that  the  sum  of  the  angles  of  a triangle  shall  equal  180°),  and 
the  more  of  these  geometrical  conditions  we  have,  the  more 
neaily  are  we  able  to  determine  what  the  true  values  of  the 
angles  are.  The  work  will  increase  in  accuracy,  therefore,  as 
the  number  of  these  conditions  increases,  and  this  is  why  sys- 
tem III.  gives  more  accurate  results  than  systems  I.  and  II. 
This  will  be  made  clear  when  the  subject  of  the  adjustment  of 
the  observations  is  considered. 

329.  The  Base-line  and  its  Connections. — The  line 
whose  length  is  actually  measured  is  called  the  base-line.  The 


* Taken  from  the  U.  S.  C.  and  G,  Survey  Report  for  1876. 


SURVEYING. 


lengths  and  distance  apart  of  sucli  lines  depend  on  the  charac- 
ter of  the  work  and  the  nature  of  the  ground.  Primary  base- 
lines are  from  three  to  ten  miles  in  length,  and  from  200  to 


Slint  Tnine>r‘ 


Morgan. 

Park 


Fig.  117.* 

600  miles  apart.  In  general,  in  primary  work,  the  distance 
apart  has  been  about  one  hundred  times  the  length  of  the 
base.  Secondary  bases  are  from  two  to  three  miles  in  length, 


Otis 


Fig.  118.* 


and  from  fifty  to  one  hundred  and  fifty  miles  apart,  the  dis- 
tance apart  being  about  fifty  times  the  length  of  base.  Ter- 
tiary bases  are  from  one  half  to  one  and  a half  miles  in  length 

* Taken  from  professional  papers,  Corps  of  Engineers  U.  S.  Army,  No.  24, 
being  the  final  report  on  the  Triangulation  of  the  United  States  Lake  Survey. 


GEODETIC  SURVEYING. 


429 


and  from  twenty-five  to  forty  miles  apart,  the  distance  apart 
being  about  twenty-five  times  the  length  of  base.* 

The  location  of  the  base  should  be  such  as  to  enable  one 
side  of  the  main  system  to  be  computed  with  the  greatest 
accuracy  and  with  the  least  number  of  auxiliary  stations  for  a 
given  length  of  base.  In  flat  open  country  the  base  may  be 
chosen  to  suit  the  location  of  the  triangulation-stations  in  the 
main  system  ; but  in  rough  country  some  of  the  main  stations 
must  often  be  chosen  to  suit  the  location  of  the  base-line.  In 
Fig.  1 17  the  location  of  the  base-line  is  almost  an  ideal  one, 
being  taken  directly  across  one  of  the  main  lines  of  the  sys- 
tem. By  referring  to  Fig.  118  it  will  be  seen  that  the  line 
Willow  Springs — Shot  Tower  is  one  of  the  fundamental  lines 
of  the  main  system,  and  the  base  is  located  directly  across  it. 
Here  the  ground  is  a flat  prairie,  and  the  base  was  chosen  to 
suit  the  stations  of  the  main  system. 

The  station  at  the  middle  of  the  base  is  inserted  in  order 
to  furnish  a check  on  the  measurements  of  the  two  portions 
as  well  as  to  increase  the  strength  of  the  system  by  increasing 
the  number  of  equations  of  conditions.  Sometimes  it  is  neces- 
sary to  use  one  or  more  auxiliary  stations  outside  the  base 
before  the  requisite  expansion  is  obtained.  Thus  suppose  the 
stations  Morgan  Park  and  Lombard  were  the  extremities  of 
the  line  of  the  main  system  whose  length  was  to  be  computed 
from  this  base,  then  the  stations  Willow  Springs  and  Shot 
Tower  might  have  been  occupied  as  auxiliary  stations  from 
which  the  line  Morgan  Park — Lombard  could  be  computed. 

330.  The  Reconnaissance. — A system  of  triangulation 
having  been  fixed  upon,  of  a given  grade  and  for  a given  pur- 


* These  intervals  between  bases  are  in  accordance  with  the  practice  that 
has  hitherto  been  followed.  The  new  method  of  measuring  base-lines  with  a 
steel  tape,  described  on  p.  450,  will  probably  change  this  practice  by  causing 
more  bases  to  be  measured,  leaving  much  shorter  intervals  to  be  covered  by 
angular  measurement. 


430 


SUR  VE  Y I NG. 


pose,  the  first  thing  to  be  done  is  to  select  the  location  of  the 
base-line  and  the  position  of  the  base-stations.  The  base  should 
be  located  on  nearly  level  ground,  and  should  be  favorably  sit- 
uated with  reference  to  the  best  location  of  the  triangulation- 
stations.  These  stations  are  then  located,  first  for  expanding 
from  the  base  to  the  main  system,  and  then  with  regard  to  the 
general  direction  in  which  the  work  is  to  be  carried,  and  to  the 
form  of  the  triangles  themselves. 

No  triangle  of  the  main  system  should  have  any  angle  less 
than  30°  nor  more  than  120°.  Although  small  angles  can  be 
measured  just  as  accurately  as  large  ones,  a given  error  in  a 
small  angle,  as  of  one  second,  has  a much  greater  effect  on  the 
resulting  distances  than  the  same  error  in  an  angle  near  90°. 
In  fact,  the  errors  in  distance  are  as  the  tabular  differences  in  a 
table  of  natural  sines,  for  given  errors  in  the  angles.  These 
tabular  differences  are  very  large  for  angles  near  0°  or  180°,  but 
reduce  to  zero  for  angles  at  90°.  The  best-proportioned  tri- 
angle is  evidently  the  equilateral  triangle,  and  the  best-propor- 
tioned quadrilateral  is  the  square.  In  making  the  reconnais- 
sance the  object  should  be  to  fulfil  these  conditions  as  nearly 
as  possible. 

The  most  favorable  ground  for  a line  of  triangles  is  a valley 
of  proper  width,  with  bald  knobs  or  peaks  on  either  side.  Sta- 
tions can  then  be  selected  giving  well-conditioned  triangles, 
with  little  or  no  clearing  out  of  lines,  and  with  low  stations. 
In  a wooded  country  the  lines  must  be  cleared  out  or  else  very 
tall  stations  must  be  used.  In  general,  both  expedients  are  re- 
sorted to.  Stations  are  built  so  as  to  avoid  the  greater  portion 
of  the  obstructions,  and  then  the  balance  is  cleared  out. 

So  much  depends  on  the  proper  selection  of  the  stations  in 
a system  of  triangulation,  as  to  time,  cost,  and  final  accuracy, 
that  the  largest  experience  and  the  maturest  judgment  should 
be  made  available  for  this  part  of  the,work.  The  form  of  the 
triangles  ; the  amount  of  cutting  necessary  to  clear  out  the 
lines  and  the  probable  resulting  damage  to  private  interests ; 


GEODETIC  SURVEYING. 


431 


the  height  and  cost  of  stations,  and  the  accessibility  of  the 
same ; the  avoidance  of  all  sources  of  atmospheric  disturbance 
on  the  connecting  lines,  as  of  factories,  lime-  or  brick-kilns,  and 
the  like,  which  might  either  obstruct  the  line  by  smoke  or  in- 
troduce unusual  refraction  from  heat ; the  freedom  from  dis- 
turbance of  the  stations  themselves  during  the  progress  of  the 
work,  and  the  subsequent  preservation  of  the  marking-stones — 
these  are  some  of  the  many  subjects  to  be  considered  in  de- 
termining the  location  of  stations. 

It  is  the  business  of  the  reconnaissance  party  not  only  to 
locate  the  stations,  but  to  determine  the  heights  of  the  same. 
A station  that  has  been  located  is  temporarily  marked  by  a 
flag  fastened  upon  a pole,  and  this  made  to  project  from  the 
top  of  a tall  tree  in  the  neighborhood.  In  selecting  a new 
station  it  is  customary  to  first  select  from  the  map  the  general 
locality  where  a station  is  needed,  and  then  examine  the  region 
for  the  highest  ground  available.  When  this  is  found,  the 
tallest  trees  are  climbed  and  the  horizon  scanned  by  the  aid  of 
a pair  of  field-glasses  to  see  if  the  other  stations  are  visible.  If 
no  tree  or  building  is  available  for  this  purpose  ladders  may 
be  spliced  together  and  raised  by  ropes  until  the  desired  height 
is  obtained. 

331.  Instrumental  Outfit. — The  reconnaissance  party  re- 
quires a convenient  means  of  measuring  angles  and  of  determ- 
ining directions  and  elevations.  For  measuring  angles  a pocket 
sextant  would  serve  very  well,  provided  the  stations  are  distinct 
or  provided  distinct  range-points  in  line  with  the  stations  may 
be  selected  by  the  aid  of  field-glasses.  A prismatic  pocket- 
compass  will  often  be  found  very  convenient  in  finding  back 
stations  which  have  been  located  and  whose  bearings  are  known. 
An  aneroid  barometer  is  desirable  for  determining  approxi- 
mate relative  elevations.  For  methods  of  using  it  in  such 
work,  see  Chapter  VI.,  p.  136.  If  to  the  above-named  instru- 
ments we  add  field-glasses,  and  creepers  for  climbing  trees,  the 
instrumental  outfit  is  fairly  complete. 


432 


SURVEYING. 


332.  The  Direction  of  Invisible  Stations. — It  often  hap- 
pens that  one  station  cannot  be  seen  from  another  on  account 
of  forest  growth,  which  may  be  cleared  out.  In  such  a case  the 
station  may  be  located  and  the  line  cleared  from  one  station  or 
from  both,  the  direction  of  the  line  having  been  determined. 
This  direction  may  always  be  computed  if  two  other  points 
can  be  found  from  each  of  wliich  both  stations  and  the  other 
auxiliary  point  are  visible.  Thus  in  Fig.  119  let  AB  be  the 

line  to  be  cleared  out,  and  let  C and  D 
be  two  points  from  which  all  the  stations 
may  be  sighted.  Measure  the  two  angles 
at  each  station  and  call  the  distance  CD 
unity.  Solve  the  triangle  BCD  for  the 
side  BC,  and  the  triangle  ADC  for  the 
side  AC.  We  now  have  in  the  triangle 
ABC  two  sides  and  the  included  angle  to 
find  the  other  angles.  When  these  are 
found  the  course  may  be  aligned  from  either  A or  B.  It  will 
often  happen  that  either  C or  D or  both  can  be  taken  at  regu- 
lar stations.  Of  course  a target  must  be  left  at  either  C or  D 
to  be  used  in  laying  out  the  line  from  A or  B.  The  above  is  a 
modification  of  the  problem  given  in  art.  1 10,  p.  107.  A use  of 
this  expedient  will  often  greatly  facilitate  the  work. 

333.  The  Heights  of  Stations  depend  on  the  relative 
heights  of  the  ground  at  the  stations  and  of  the  intervening 
region.  If  the  surface  is  level,  then  the  heights  of  stations 
depend  only  on  their  distance  apart.  In  any  case  the  dis- 
tance apart  is  so  important  a function  of  the  necessary  height 
that  it  is  well  to  know  what  the  heights  would  have  to  be  for 
level,  open  country. 

The  following  table*  gives  the  height  of  one  station  when 
the  other  is  at  the  ground  level,  for  open,  level  country: 


* Taken  from  Report  of  U.  S.  Coast  and  Geodetic  Survey  for  1882. 


GEODETIC  SURVEYING. 


433 


DIFFERENCE  IN  FEET  BETWEEN  THE  APPARENT  AND  TRUE 
LEVEL  AT  DISTANCES  VARYING  FROM  i TO  66  MILES. 


Dis- 

tance, 

miles. 

Difference  in  feet  for — 

Dis- 

tance, 

miles. 

Difference  in  feet  for — 

Curvature. 

Refraction. 

Curvature 

and 

Refraction. 

Curvature. 

Refraction. 

Curvature 

and 

Refraction. 

I 

0.7 

0. 1 

0.6 

34 

771-3 

108.0 

663.3 

2 

2.7 

0,4 

2.3 

35 

817.4 

II4.4 

703.0 

3 

6.0 

0.8 

5-2 

36 

864.8 

I2I.  I 

743-7 

4 

10.7 

15 

9.2 

37 

913-5 

127.9 

785.6 

5 

16.7 

2-3 

14.4 

38 

963.5 

134.9 

828.6 

6 

24.0 

3-4 

20.6 

39 

1014. 9 

142. 1 

872.8 

7 

32.7 

4.6 

28.1 

40 

1067.6 

149-5 

918.1 

8 

42.7 

6.0 

36-7 

41 

1121.7 

157.0 

964.7 

9 

54-0 

7-6 

46.4 

42 

1177. 0 

164.8 

1012. 2 

10 

66.7 

9-3 

57-4 

43 

1233.7 

172.7 

1061. 0 

II 

80.7 

ir-3 

69-4 

44 

1291.8 

180.8 

IIII.O 

12 

96.1 

13-4 

82.7 

45 

1351. 2 

189*.  2 

1162.0 

13 

112.8 

15.8 

97-0 

46 

1411.9 

197.7 

1214. 2 

14 

130.8 

18.3 

112.5 

47 

1474.0 

206.3 

1267.7 

15 

150. 1 

21.0 

129. 1 

48 

1537.3 

215.2 

1322. I 

16 

170.8 

23-9 

146.9 

49 

1602.0 

224.3 

^377-7 

17 

192.8 

27.0 

165.8 

50 

1668 . I 

233.5 

1434.6 

18 

216.2 

30.3 

185.9 

51 

1735.5 

243-0 

1492.5 

19 

240.9 

33-7 

207.2 

52 

1804,2 

252.6 

1551.6 

20 

266.9 

37-4 

229.5 

53 

1874.3 

262.4 

1611.9 

21 

94-3 

41.2 

253-1 

54 

1945.7 

272.4 

1673-3 

22 

322.9 

45-2 

277-7 

55 

2018.4 

282.6 

1735.8 

23 

353-0 

49-4 

303.6 

56 

2092.5 

292.9 

1799.6 

24 

384-3 

53-8 

330.5 

57 

2167.9 

303.5 

1864.4 

25 

417.0 

58.4 

358.6 

58 

2244.6 

314.2 

1930.4 

26 

451. 1 

63.1 

388.0 

59 

2322.7 

325.2 

1997-5 

27 

486.4 

68.1 

418.3 

60 

2402 . I 

336.3 

2065.8 

28 

523-1 

73-2 

449.9 

61 

2482,8 

347.6 

2135.2 

29 

561.2 

78.6 

482.6 

62 

2564.9 

359-1 

2205.8 

30 

600.5 

84.1 

516.4 

63 

2648.3 

370.8 

2277-5 

31 

641.2 

89.8 

551-4 

64 

2733.0 

382.6 

2350.4 

32 

683.3 

95-7 

587.6 

65 

2819.1 

394.7 

2424.4 

33 

726.6 

loi  .7 

624.9 

66 

2906 . 5 

406.9 

2499.6 

28 


434 


SURVEYING. 


square  of  distance 

urvature  _ diameter  of  earth  ’ 


Log  curvature  = log  square  of  distance  in  feet  — 7.6209807 ; 


Refraction  = where  K represents  the  distance  in  feet, 


R the  mean  radius  of  the  earth  (log  R = 7*3199507),  and  m the 
coefficient  of  refraction,*  assumed  at  .070,  its  mean  value,  sea- 
coast  and  interior. 


Curvature  and  refraction  = (i  — 2w) 


Or,  calling  h the  height  in  feet,  and  K the  distance  in  statute 
miles,  at  which  a line  from  the  height  h touches  the  horizon, 
taking  into  account  terrestrial  refraction,  assumed  to  be  of  the 
same  value  as  in  the  above  table  (.070),  we  have 


\ni 

•7575’ 


1.7426* 


The  following  examples  will  serve  to  illustrate  the  use  of 
the  preceding  table  : 

I.  Elevation  of  Instrument  required  to  overcome  Curvature 
and  Refraction. — Let  us  suppose  that  a line,  A to  B,  was  18 
miles  in  length  over  a plain,  and  that  the  instrument  could  be 
elevated  at  either  station,  by  means  of  a portable  tripod,  to  a 
height  of  20  or  30  or  50  feet.  If  we  determine  upon  36.7  feet 
at  A,  the  tangent  would  strike  the  curve  at  the  distance  rep- 
resented by  that  height  in  the  table,  viz.,  8 miles,  leaving  the 
curvature  (decreased  by  the  ordinary  refraction)  of  10  miles  to 
be  overcome.  Opposite  to  10  miles  we  find  57.4  feet,  and  a 


* See  discussion  on  refraction,  under  Geodetic  Levelling,  this  chapter. 


GEODETIC  SURVEYING. 


435 


signal  at  that  height  erected  at  B would,  under  favorable 
refraction,  be  just  visible  from  the  top  of  the  tripod  at  or 
be  on  the  same  apparent  level.  If  we  now  add  8 feet  to  tripod 
and  8 feet  to  signal-pole,  the  visual  ray  would  certainly  pass  6 
feet  above  the  tangent  point,  and  20  feet  of  the  pole  would  be 
visible  from  A. 

11.  Elevations  required  at  given  Distances. — If  it  is  desired 
to  ascertain  whether  two  points  in  the  reconnaissance,  esti- 
mated to  be  44  miles  apart,  would  be  visible  one  from  the 
other,  both  elevations  must  be  at  least  278  feet  above  mean 
tide,  or  one  230  feet  and  the  other  331  feet,  etc.  This  sup- 
poses that  the  intervening  country  is  low,  and  that  the  ground 
at  the  tangent  point  is  not  above  the  mean  surface  of  the 
sphere.  If  the  height  of  the  ground  at  this  point  should  be 
200  feet  above  mean  tide,  then  the  natural  elevations  should 
be  478  or  430  and  531  feet,  etc.,  in  height,  and  the  line  is 
barely  possible.  To  insure  success,  the  theodolite  must  be 
elevated  at  both  stations  to  avoid  high  signals. 

Since  the  height  of  station  increases  as  the  square  of  the 
distance,  it  is  evident  that  the  minimum  aggregate  station 
height  is  obtained  by  making  them  of  equal  height.  Or,  if 
the  natural  ground  is  higher  at  one  station  than  the  other, 
then  the  higher  station  should  be  put  on  the  lower  ground — 
that  is,  when  the  intervening  country  is  level.  If,  however, 
the  obstruction  is  due  to  an  intervening  elevation,  the  higher 
station  should  be  the  one  nearer  the  obstruction. 

Sometimes  a very  high  degree  of  refraction  is  utilized  to 
make  a connection  on  long  lines.  Thus  on  the  primary  trian- 
gulation of  the  Great  Lakes  three  lines  respectively  100,  93, 
and  92  miles  in  length  were  observed  across  Lake  Superior, 
which  could  not  have  been  done  except  that  the  refraction  was 
found  sometimes  to  exceed  twice  its  average  amount.  The  line 
from  station  Vulcan,  on  Keweenaw  Point,  to  station  Tip-Top 
in  Canada,  was  100  miles  in  length.  The  ground  at  statior, 


436 


SUItVEVINC. 


Vulcan  was  726  feet  above  the  lake,  and  the  observing  station 
was  elevated  75  feet  higher,  making  801  feet  above  the  surface 
of  the  lake.  The  station  at  Tip-Top  was  1523  feet  above  the 
lake,  the  observing  tripod  being  only  3 feet  high.  From  the 
above  table  we  find  that  the  line  of  sight  from  Vulcan  would 
become  tangent  to  the  surface  of  the  lake  at  a distance  of  37.4 
miles,  and  that  from  Tip-Top  at  a distance  of  51.5  miles,  thus 
leaving  a gap  of  about  eleven  miles  between  the  points  of 
tangency,  for  ordinary  values  of  the  refraction.  If  this  inter- 
val were  equally  divided  between  the  two  stations  and  these 
raised  to  the  requisite  height,  we  would  find  from  the  table 
that  Tip-Top  would  have  to  be  elevated  some  340  feet  and 
Vulcan  some  260  feet.  Since  this  was  not  done,  we  must  con- 
clude that  an  occasional  excessive  value  of  the  refraction  was 
sufficient  to  bend  these  rays  of  light  by  about  these  amounts 
in  addition  to  the  ordinary  curvature  from  this  source.  In 
other  words,  the  actual  refraction  when  one  of  these  stations 
was  visible  from  the  other  must  have  been  more  than  double 
its  mean  amount. 

The  following  is  a synopsis  of  the  heights  of  the  stations 
built  for  the  observation  of  horizontal  angles  in  the  primary 
triangulation  of  the  Great  Lakes: 


Total  number  of  stations  * 243 

Combined  height  of  stations 14,100  feet 

Average  height  of  stations 58"  “ 

Average  height  of  stations  from  Chicago  to  Buffalo 81.3  “ 

Number  of  stations  less  than  10  feet  high 22 

“ “ from  10  feet  to  24  feet  in  height 18 

“ “ “ 25  “ 49  “ “ 50 

“ “ “ 50  “ 74  “ “ 71 

“ “ “75  “ 99  “ “ 47 

“ “ “ 100  “ 109  “ “ 18 

“ “ “ no  “ 119  “ “ 15 

“ “ “ 120  “ 124  “ “ 2 


*Only  stations  built  expressly  for  the  work  are  here  included.  Sometimes 
buildings  or  towers  were  utilized  in  addition  to  these. 


GEODETIC  SURVEYING. 


437 


The  heights  above  given  are  the  heights  at  which  the  in- 
strument was  located  above  the  ground.  The  targets  were 
usually  elevated  from  5 to  30  feet  higher. 

The  excessive  heights  of  the  stations  from  Chicago  to 
Buffalo  are  due  to  the  country  being  very  heavily  timbered, 
and  the  surface  only  gently  rolling.  In  the  vicinity  of  Lake 
Superior  they  averaged  only  about  35  feet  high,  while  from 
Buffalo  to  the  eastern  end  of  Lake  Ontario  they  averaged  51 
feet  in  height. 

334.  Construction  of  Stations. — If  it  is  found  necessaiy 
to  build  tall  stations,  two  entirely  separate  structures  must  be 


SCAFFOLD  OBSERVING  TRIPOD 

Fig.  120. 


erected,  one  for  carrying  the  instrument  and  one  for  sustain- 
ing the  platform  on  which  the  observer  stands.  These  should 
have  no  rigid  connection  with  each  other.  These  structures  are 
shown  in  plan  and  elevation  in  Figs.  120  and  121.  The  inner 
station  is  a tripod  on  which  the  instrument  rests;  this  is  sur- 
rounded by  a quadrangular  structure,  shown  separately  in  ele- 
vation to  prevent  confusion.  Both  structures  are  built  entirely 
of  wood,  the  outer  one  being  usually  carried  up  higher  than 


438 


SURVEYING. 


the  tripod  (not  shown  in  the  drawinp^),  and  the  target  fixed  to 
its  apex.  This  upper  framework  serves  also  to  support  an 
awning  to  shade  the  instrument  from  the  sun.  For  lower  sta- 
tions a simpler  construction  will  serve,  but  the  observer’s  plat- 
form must  in  all  cases  be  separate  from  the  instrument  tripod. 
The  wire  guys  and  wooden  braces  shown  in  Fig.  120  were  not 
used  on  the  U.  S.  Lake  Survey  stations. 

For  stations  less  than  about  15  feet  in  height  the  design 


■-©- 


/i 


/'  |\'\ 

"><  \\ 

/ '''-1  \ 

'"''4  \ 

\ 

I ! 

AWt i 1'4 

GROUND  PLAN  Scale  ^ 

Fig.  121.  ' 


shown  in  Figs.  122  and  123  may  be  used.  Here  the  outer 
platform  on  which  the  observer  stands  is  entirely  separate  from 
the  tripod  which  supports  the  instrument.  For  ground  stations 
a post  firmly  planted  serves  very  well,  or  a tree  cut  off  to  the 
proper  height.  The  common  instrument  tripod  will  seldom  be 
found  satisfactory  for  good  work.  Sometimes  extra  heavy  and 
stable  tripods  of  the  ordinary  pattern  have  given  excellent  re- 
sults. 

335.  Targets. — The  requisites  of  a good  target  are  that  it 
shall  be  clearly  visible  against  all  backgrounds,  readily  bisected, 


GEODETIC  SURVEYING. 


439 


rigid,  capable  of  being  accurately  centred  over  the  station,  and 
so  constructed  that  the  centre  of  the  visible  portion,  whether 
in  sun  or  in  shade,  shall  coincide  with  its  vertical  axis. 


It  is  not  easy  always  to  fulfil  these  conditions  satisfactorily. 
To  make  it  visible  against  light  or  dark  backgrounds,  it  is  well 


to  paint  it  in  alternating  black  and  white  belts.  For  ready  bi- 
section it  should  be  as  narrow  as  possible  for  distinctness.  This 


440 


SURVEYING. 


is  accomplished  by  making  the  width  subtend  an  angle  of  from 
two  to  four  seconds  of  arc.  Since  the  arc  of  one  second  is 
three  tenths  of  an  inch  for  one-mile  radius,  an  angle  of  four 
seconds  would  give  a target  one  tenth  of  a foot  in  diameter  for 
one-mile  distances,  or  one  foot  in  diameter  for  ten-mile  dis- 
tances. Something  depends  on  the  magnifying  power  of 
the  telescope  used.  The  design  shown  in  Fig.  124  will  satis- 


Fig.  124. 


factorily  satisfy  the  conditions  as  to  rigidity  and  convenience 
of  centering.  Of  course  it  should  stand  vertically  over  the 
station  so  that  a reading  could  be  taken  on  any  part  of  its 
height.  The  last  condition  is  not  so  easily  satisfied.  If  a 
cylinder  or  cone  be  used  the  illuminated  portion  only  will 
appear  when  the  sun  is  shining,  and  a bisection  on  this  portion 
may  be  several  inches  to  one  side  of  the  true  axis. 


GEODETIC  SURVEYING. 


441 


The  target  is  then  said  to  present  a phase,  and  corrections 
for  this  are  sometimes  introduced.  It  is  much  better,  however, 
to  use  a target  which  has  no  phase.  If  the  target  is  to  be  read 
mostly  from  one  general  direction,  a surface,  as  a board,  may 
be  used  ; but  if  the  target  is  to  be  viewed  from  various  points 
of  the  compass,  then  from  those  stations  which  lie  nearly  in  the 
plane  of  the  target  it  would  not  be  visible,  from  its  width  being 
so  greatly  foreshortened. 

In  this  case  two  planes  could  be  set  at  right  angles,  one 
above  the  other.  One  or  both  would  then  be  visible  from  all 
points,  and  since  their  axes  are  coincident,  either  one  could  be 
used.  The  objection  to  this  would  be  that  the  upper  disk  would 
cast  its  shadow  at  times  on  the  lower  one,  leaving  one  side  in 
sun  and  the  other  in  shade,  thus  giving  rise  to  the  very  evil  it 
is  sought  to  eliminate.  A very  satisfactory  solution  of  this 
problem  was  made  on  the  Mississippi  River  Survey  by  means 
of  the  following  device  (Fig.  125):  Four  galvanized-iron  wires, 
about  three-sixteenths  inch  in  diameter,  are  bent  into  a circle  of, 
say,  four  inches  in  diameter,  and  soldered.  To  these  four  circles 
are  attached  four  vertical  wires  about  one  fourth  inch  in  diam- 
eter and  four  feet  long,  as  shown  in  the  accompanying  figure. 
All  joints  to  be  securely  soldered,  the  size  of  the  wire  increas* 
ing  with  the  size  of  the  target.  The  target  is  now  divided  into 
a number  of  zones  by  stretching  black  and  white  canvas  alter- 
nately and  in  opposite  ways  between  the  opposing  uprights, 
making  diametral  sections.  If  there  are  more  than  two  zones, 
those  marked  by  the  same  color  should  have  the  canvas  cross- 
ing in  different  ways,  so  that  if  one  plane  is  nearly  parallel  to 
any  line  of  sight  the  other  plane  of  this  color  will  be  nearly  at 
right  angles  to  it.  This  target  has  no  phase,  is  visible  against 
any  background,  and  readily  mounted.  A wooden  block  may 
be  inserted  at  bottom,  with  a hole  in  the  axis  of  the  target. 
This  may  then  be  set  over  a nail  marking  the  station.  The 
target  is  held  at  top  by  wire  guys  leading  off  to  stakes  in  the 


442 


SUR  VE  YING. 


ground.  Such  a target  could  be  mounted  on  top  of  the  pole 
shown  in  Fig.  124,  if  it  should  be  found  necessary  to  elevate  it. 

336.  Heliotropes. — When  the  distance  between  stations 
is  such  that,  owing  to  the  distance,  the  state  of  the  atmosphere, 
or  the  small  size  of  the  objective  used,  a target  would  appear 
indistinct,  or  perhaps  not  be  visible  at  all,  the  reflected  rays  of 
the  sun  may  be  made  to  serve  in  place  of  a target.  This  limit- 
ing distance  is  usually  about  twenty  miles.  Any  device  for 
accomplishing  this  purpose  may  be  called  a heliotrope.  In 
Figs.  126  and  127  are  two  forms  of  such  an  instrument.  That 


shown  in  Fig.  126  is  a telescope  mounted  with  a vertical  and 
horizontal  motion.  This  is  turned  upon  the  station  occupied 
by  the  observer,  and  is  then  left  undisturbed.  On  the  tele- 
scope are  mounted  a mirror  and  two  disks*  with  circular  open- 
ings. The  mirror  has  two  motions  so  that  it  can  be  put  into 
any  position.  Its  centre  is  coincident  with  the  axis  of  the 
disks,  in  all  positions.  The  mirror  may  be  turned  so  as  to 


* The  disk  next  to  the  mirror  is  unnecessary. 


GEODETIC  SURVEYING. 


443 


throw  a beam  of  light  symmetrically  through  the  forward  disk, 
in  which  position  the  reflected  rays  are  parallel  to  the  axis  of 
the  telescope,  and  hence  fall  upon  the  distant  point. 

The  heliotrope  shown  in  Fig.  127  is  to  be  used  in  conjunc- 
tion with  a single  disk,  which  may  be  a plain  board  mounted 
on  a plank  with  the  mirror.  The  silvering  is  removed  from  a 
small  circle  at  the  centre  of  the  mirror.  The  disk  has  a small 
hole  through  it  as  high  above  its  base  as  the  clear  space  on  the 
mirror  is  above  the  plank.  The  operator  points  the  apparatus 
by  sighting,  through  the  clear  spot  on  the  mirror  and  the  open- 
ing in  the  disk,  to  the  distant  station.  If  the  plank  be  fas- 
tened in  this  position  the  attendant  now  has  only  to  move  the 
mirror  so  as  to  keep  the  cone  of  reflected  rays  symmetrically 
covering  the  opening  in  the  disk,  and  the  light  will  be  thrown 
to  the  distant  station. 

Since  the  cone  of  incident  rays  subtends  an  angle  of  about 
thirty-two  minutes,  the  cone  of  reflected  rays  subtends  the 
same  angle.  The  base  of  this  cone  has  a breadth  of  about 
fifty  feet  to  the  mile  distance,  or  at  a distance  of  twenty  miles 
the  station  sending  the  reflection  is  visible  over  an  area  in  a 
vertical  plane  1000  feet  in  diameter.  The  alignment  of  the 
heliotrope  need  not,  therefore,  be  very  accurate.  This  align- 
ment may  vary  as  much  as  fifteen  minutes  of  arc  on  either  side 
of  the  true  line.  This  is  nearly  o.oi  of  a foot  in  a distance  of 
two  feet.  If  the  bearing,  or  direction,  of  the  distant  station  is 
once  determined,  it  may  be  marked  on  the  station  by  some 
means  within  this  limit,  and  a very  rude  contrivance  used  for 
sending  the  reflected  ray,  or  flash,  as  it  is  called.  Thus,  a mir- 
ror and  a disk  with  the  requisite  movements  may  be  mounted 
on  the  ends  of  a board  or  pole  from  five  to  twenty  feet  long, 
and  when  this  is  properly  aligned  it  serves  as  well  as  any  other 
more  expensive  apparatus.  The  hole  in  the  disk  should  usually 
subtend  an  angle  at  the  observer’s  station  of  something  less 
than  one  second  of  arc,  which  is  a width  of  three-tenths  of  an 


444 


SURVEYING. 


inch  to  the  mile  distance.  On  the  best  work  with  large  instru- 
ments it  should  subtend  an  angle  of  less  than  one  half  a 
second,  the  minimum  effective  opening  depending  almost 
wholly  on  the  condition  of  the  atmosphere.^' 

Whatever  form  of  heliotrope  is  used,  an  attendant  is  re- 
quired to  operate  the  apparatus.  Evidently  it  can  be  used 
only  on  clear  days,  whereas  cloudy  weather  is  much  better 
adapted  to  this  kind  of  work,  since  the  atmosphere  then  trans- 
mits so  much  clearer  and  steadier  an  image. 

The  heliotrope  can  be  used  as  a means  of  communication 
between  distant  stations  by  some  fixed  code  of  flashing  sig- 
nals, and  it  has  been  so  used  very  often  with  great  advantage 
to  the  work.  The  attendant  on  the  heliotrope,  usually  called 
a flasher,  can  thus  know  when  the  observer  is  reading  his  sig- 
nals, when  he  is  through  at  that  station,  and,  in  general,  can  re- 
ceive his  instructions  from  his  chief  direct  from  the  distant 
station. 

337.  Station  Marks. — If  the  triangulation  is  to  serve  for 
the  fixing  of  points  for  future  reference,  then  these  points  must 
be  marked  in  some  more  or  less  permanent  manner.  In  this 
case  the  station  has  been  chosen  with  this  in  view,  so  that  if 
possible  it  has  been  provided  that  even  the  surface  for  a few 
feet  around  the  station  shall  remain  undisturbed.  To  insure 
against  disturbance  from  frost  or  otherwise,  the  real  mark  is 
usually  set  several  feet  underground.  Many  different  means 
are  employed  to  mark  these  points.  The  underground  mark 
is  to  serve  only  when  the  superficial  marks  have  been  dis- 
turbed, there  being  always  left  a mark  of  some  kind  projecting 
above  ground.  On  the  U.  S.  Lake  Survey,  “ the  geodetic 
point  is  the  centre  of  a J-inch  hole  drilled  in  the  top  of  a stone 

* Reflected  sunlight  has  been  seen  a distance  of  sixty  miles,  through  an 
opening  one  inch  in  diameter,  which  then  subtended  an  angle  of  but  one  eigh- 
teenth of  one  socond  of  arc  at  the  instrument.  This  would  require  a very  clear 
atmospliere. 


GEODETIC  SURVEYING. 


445 


two  feet  by  six  inches  by  six  inches,  sunk  two  and  one-half 
feet  below  the  surface  of  the  ground.  When  the  occupation 
of  the  station  is  finished,  a second  stone  post,  rising  eight 
inches  above  the  ground,  is  placed  over  the  first  stone.  Three 
stone  reference-posts,  three  feet  long,  rising  about  a foot  above 
the  ground,  are  set  within  a few  hundred  feet  of  the  station, 
where  they  are  the  least  likely  to  be  disturbed.  A sketch  of 
the  topography  within  a radius  of  400  metres  about  the  sta- 
tion is  made,  and  the  distances  and  azimuths  of  the  reference- 
marks  are  accurately  determined.” 

When  the  station  is  located  in  natural  rock  a copper  bolt 
may  be  set  to  mark  the  geodetic  point. 

On  the  Mississippi  River  survey,  stations  had  to  be  set  on 
ground  subject  to  overflow.  These  were  to  serve  both  for 
geodetic  points  and  for  bench-marks,  both  their  geographical 
position  and  their  elevation  being  accurately  determined. 
Both  the  rank  growth  and  the  sedimentary  deposits  from  the 
annual  overflows  would  soon  obliterate  any  mark  which  was 
but  slightly  raised  above  the  surface.  After  much  study  given 
to  the  subject,  the  following  method  of  marking  such  points 
was  adopted : A flat  stone  eighteen  inches  square  and  four 
inches  thick,  dressed  on  the  upper  side,  has  a hole  drilled  in 
the  centre,  into  which  a copper  bolt  is  leaded,  the  end  project- 
ing a quarter  of  an  inch  above  the  face  of  the  stone.  The 
U S 

stone  is  marked  thus,  ^ and  is  placed  three  feet  under 

ground.  On  this  stone,  and  centred  over  the  copper  bolt,  a 
cast-iron  pipe  four  inches  in  diameter  and  five  feet  long  is 
placed,  and  the  dirt  tamped  in  around  it.  The  pipe  is  large 
enough  to  admit  a levelling-rod.  The  top  is  closed  with  a cap, 
which  is  fastened  to  the  pipe  by  means  of  a bolt.  The  eleva- 
tions of  both  the  top  of  the  pipe  and  of  the  stone  are  de- 
termined. 


Fig.  128.— Four-Metre  Contact-slide  Base  Apparatus,  after  the  Design  of  Prof.  J.  E.  Hilgard,  Superintendent  U.  S.  C.  and  G.  Survey. 


GEODETIC  SURVEYING. 


447 


MEASUREMENT  OF  THE  BASE-LINE. 

338.  Methods. — The  methods  heretofore  employed  in  meas- 
uring a base-line  have  depended  on  the  degree  of  accuracy 
requisite.  If  an  accuracy  of  one  in  one  million  was  desired,  then 
the  most  elaborate  primary  apparatus  has  been  used,  such  as 
may  be  found  described  in  the  U.  S.  Coast  and  Geodetic  Survey 
Reports  for  1873  and  1882,  or  in  the  Primary  Triangulation  of 
the  U.  S.  Lake  Survey.*  For  an  accuracy  of  one  in  fifty 
thousand  or  one  in  one  hundred  thousand,  more  simple  appli- 
ances have  been  used,  such  as  that  shown  in  Fig.  128.  This 
apparatus  is  fully  described  and  illustrated  in  the  U.  S. 
Coast  and  Geodetic  Survey  Report  for  1880,  Appendix 
No.  17.  It  consists  essentially  of  a four-metre  steel  bar, 
with  zinc  tubes  on  either  side  of  it.  One  of  these  zinc  tubes 
is  attached  to  the  steel  bar  at  one  end  and  the  other  at  the 
other  end.  Since  the  expansion  of  zinc  is  about  two  and  a 
half  times  that  of  steel,  it  is  evident  that  the  corresponding 
ends  of  the  zinc  bars  will  have  a relative  motion  with  reference 
to  each  other  as  the  temperature  changes.  This  relative  mo- 
tion is  observed  by  means  of  the  vernier  scales  attached  to  the 
ends  of  the  zinc  tubes.  When  the  absolute  length  of  the  steel 
bar  and  the  coefficients  of  expansion  of  both  the  steel  and  zinc 
bars  are  determined,  and  the  readings  of  the  vernier  scales  for 
a given  temperature,  then  any  other  temperature  will  be  in- 
dicated by  the  scale-readings.  This  combination  thus  becomes 
a metallic  thermometer,  from  which  the  temperature  of  the 
steel  rod  may  be  accurately  known  while  in  use  in  the  field. 
This  assumes  that  the  steel  and  zinc  rods  are  at  the  same 
temperature  at  all  times,  and  that  the  changes  in  length  due  to 
changes  in  temperature  occur  simultaneously  with  the  tempera- 


*This  is  a large  quarto  volume  of  920  pp.  and  30  plates,  describing  the 
methods  and  results  of  the  geodetic  work  of  the  U.  S.  Lake  Survey.  It  is  a 
most  valuable  contribution  to  the  science  of  geodesy,  and  is  No.  24  of  the 
Professional  Papers  of  the  Corps  of  Engineers  of  the  U.  S.  Army,  1882. 


448 


S UR  VE  YING. 


turc  changes.  Unfortunately,  this  latter  condition  is  not  ful- 
filled in  the  case  of  zinc. 

From  elaborate  observations  on  the  relative  expansions  of 
steel  and  zinc  bars  on  the  United  States  Lake  Survey,  it  was 
found  that  zinc  is  like  glass  in  that  its  volume-change  is  not 
wholly  coincident  with  its  corresponding  temperature-change, 
a residual  portion  of  its  change  of  volume  requiring  a consid- 
erable time  for  its  completion.  In  other  words,  the  volume- 
change  lags  behind  the  temperature-change,  so  that  its  volume 
is  not  truly  indicated  by  its  temperature,  it  being  rather  a 
function  of  the  changes  in  temperature  for  an  indefinite  pre- 
vious period.  Zinc  is,  therefore,  not  a fit  metal  to  use  in  the 
most  accurate  measurements,  although  it  is  sufficiently  reliable 
for  a secondary  apparatus. 

When  two  combinations  of  bars  described  above  are  prop- 
erly protected  from  sun,  wind,  and  from  too  sudden  and  varia- 
ble temperature-changes,  and  when  they  are  mounted  in  such 
a way  as  to  enable  them  to  be  aligned  both  horizontally  and 
vertically,  with  suitable  provision  for  making  exact  contacts 
between  the  ends  of  the  steel  bars,  they  then  form  a base 
apparatus. 

Sometimes  simple  wooden  or  iron  rods  have  been  used  in 
this  way,  but  then  the  great  source  of  error  is  in  not  knowing 
the  mean  temperature  (and  hence  length)  of  the  rods  at  any 
time.  If  mercurial  thermometers  are  used,  these  may  be  many 
degrees  warmer  or  cooler  than  the  bar,  since  the  mercury  bulb 
is  so  much  smaller  in  cross-section  than  the  bar,  and  therefore 
responds  more  quickly  to  changes  in  temperature.  The  steel- 
zinc  combination  is  an  ideal  one,  and  would  be  practically  per- 
fect if  zinc  were  as  reliable  a metal  as  steel.  The  best  metals 
for  metallic  thermometers  are  probably  steel  and  brass,  the 
coefficient  of  expansion  of  the  latter  being  about  1.5  times  the 
former.* 


* Mr.  E.  S.  Wheeler,  U.  S.  Asst.  Engr.,  who  has  had  a very  large  experi- 
perience  in  the  measurement  of  primary  base-lines  on  the  U.  S.  Lake  Survey, 


GEODETIC  SURVEYING. 


449 


The  Steel  Tape  furnishes  the  most  convenient,  rapid,  and 
economical  means  for  measuring  any  distance  for  any  desired 
degree  of  accuracy  up  to  about  one  in  three  hundred  thousand, 
and  if  the  most  favorable  times  are  chosen,  an  accuracy  of  i 
in  1,000,000  may  be  attained.  It  is  probable,  therefore,  that 
all  engineering  measurements,  even  including  primary  base- 
lines, will  yet  be  made  by  the  steel  tape  or  by  steel  and  brass 
wires.  The  conditions  of  use  depend  on  the  accuracy  re- 
quired. Let  us  suppose  the  absolute  length,  coefficient  of 
expansion,  and  modulus  of  elasticity  have  been  accurately 
determined.  Any  distance  can  then  be  measured  in  absolute 
units  within  an  accuracy  of  one  in  one  million,  by  taking  due 
precautions  as  to  temperature  and  mechanical  conditions. 
The  length  of  the  tape  for  city  work  is  usually  fifty  feet,  and 
its  cross-section  about  \ inch  by  A inch.  That  used  in  New 
York  City  is  inch  wide  by-^^^  inch  thick.  For  mining,  topo- 
graphical, and  railroad  surveying  a length  of  one  hundred  feet, 
with  a cross-section  of  about  \ by  inch,  is  most  convenient. 
For  base-line  measurement  the  length  should  be  from  three 
hundred  to  five  hundred  feet,  and  its  cross-section  from  two  to 
three  one-thousandths  of  a square  inch.  For  an  accuracy  of 
one  in  five  thousand  the  tape  may  be  used  in  all  kinds  of 
weather,  held  and  stretched  by  hand,  the  horizontal  position 
and  amount  of  pull  estimated  by  the  chainmen.  The  tempera- 
ture may  be  estimated,  or  read  from  a thermometer  carried 
along  for  the  purpose.  On  uneven  ground,  the  end  marks  are 
given  by  plumb-line. 

For  an  accuracy  of  one  in  fifty  thousand  the  mean  tem- 
perature of  the  tape  should  be  known  to  the  nearest  degree 
Fahrenheit,  the  slope  should  be  determined  by  stretching  over 
stakes,  or  on  ground  whose  slope  is  determined,  and  the  pull 


recommends  the  use  of  a single  bar  packed  in  ice,  with  micrometer  microscopes 
mounted  on  iron  stands  to  mark  the  end  positions  of  the  bar.  By  this  means 
a constant  length  of  standard  can  be  obtained.  This  has  never  yet  been  done, 
however. 


29 


450 


SURVEYING. 


should  be  measured  by  spring  balances.  The  work  could  then 
be  done  in  almost  any  kind  of  cloudy  weather.  For  an  accu- 
racy of  one  in  five  hundred  thousand,  extreme  precautions 
must  be  taken.  The  mean  temperature  must  be  determined 
to  about  one  fifth  of  a degree  F.,  the  slope  must  be  accurately 
determined  by  passing  the  tape  over  points  whose  elevations 
above  a given  datum  are  known,  the  pull  must  be  known  to 
within  a few  ounces,  and  all  friction  must  be  eliminated.  The 
largest  source  of  error  is  apt  to  be  the  temperature.  On  clear 
days,  the  temperature  of  the  air  varies  rapidly  for  varying 
heights  above  the  ground,  and,  besides,  the  temperature  of  the 
tape  would  neither  be  that  of  the  air  surrounding  it,  nor  of  the 
bulb  of  a mercurial  thermometer.  In  fact,  there  is  no  way  of 
determining  by  mercurial  thermometer,  even  within  a few 
degrees,  the  mean  temperature  of  a steel  tape  lying  in  the  sun, 
either  on  or  at  varying  heights  above  the  ground.  The  work 
must  then  be  done  in  cloudy  weather,  and  when  air  and  ground 
are  at  about  the  same  temperature. 

There  should  also  be  no  appreciable  wind,  both  on  account 
of  its  mechanical  action  on  the  tape,  and  from  the  temperature- 
variations  resulting  therefrom. 

339.  Method  of  Mounting  and  Stretching  the  Tape. — 
To  eliminate  all  friction,  the  tape  is  suspended  in  hooks  about 
two  inches  long,  these  being  hung  from  nails  in  the  sides  of 
“ line-stakes"  driven  with  their  front  edges  on  line.  These 
stakes  may  be  from  twenty  to  one  hundred  feet  apart.  The 
nails  may  be  set  on  grade  or  not,  as  desired  ; but  if  not  on 
grade,  then  each  point  of  support  must  have  its  elevation  deter- 
mined. A low  point  should  not  intervene  between  two  higher 
ones,  or  the  pull  on  the  tape  may  lift  it  from  this  support. 
“ Marking-stakes"  are  set  on  line  with  their  tops  about  two  feet 
above  ground,  at  distances  apart  equal  to  a tape-length,  say  300 
feet.  Zinc  strips  about  one  and  one  half  inches  wide  are  tacked 
to  the  tops  of  these  stakes,  and  on  these  the  tape-lengths  are 


GEODETIC  SURVEYING. 


451 


marked  with  a steel  point.  These  strips  remain  undisturbed 
until  all  the  measurements  are  completed,  when  they  n 
preserved  for  future  reference.  In  front  of  the  marking-stake 
three  “ table-stakes”  are  driven,  on  which  to  rest  the  stretching 
apparatus,  and  in  the  rear  a “ straining-stake”  to  which  to  at- 
tach the  rear  end  of  the  tape.  These  auxiliary  stakes  are  set 
two  or  three  feet  away  from  the  marking-stake,  and  enough 


lower  to  bring  the  tape,  when  stretched,  to  rest  on  the  top 
of  the  marking-stake. 

The  stretching  apparatus  is  shown  in  Fig.  129.*  A chain 
is  attached  to  the  end  of  the  tape,  and  this  is  hooked  over  the 


* This  figure,  and  the  method  here  described,  are  taken  from  the  advance- 
sheets  of  the  Report  of  the  Missouri  River  Commission  for  1886.  The  work 
was  in  charge  of  Mr.  O.  B.  Wheeler,  U.  S.  Asst.  Engr.,  who  first  used  this 
method  on  the  Missouri  River  Survey  in  1885.  The  author  had  previously 
developed  and  used  the  general  method,  except  that  he  stretched  his  tape  by  a 
weight  hung  by  a line  passing  through  a loop  which  was  kept  at  an  angle  of 
45“  with  the  vertical,  and  his  end  marks  were  made  on  copper  tacks  driven  into 
the  tops  of  the  stakes.  He  had  also  used  spring  balances  for  stretching  the  tape. 


452 


SURVEYING. 


staple  K which  is  attaclied  to  the  block  KHK' . This  block  is 
hinged  on  a knife-edge  at  //,  and  is  weighed  at  K'  by  the  load 
P,  The  hinge  bearing  at  H is  attached  to  a slide  which  is 
moved  by  the  screw  S working  in  the  nut  N.  The  whole  ap- 
paratus is  set  on  the  three  table-stakes  in  front  of  the  marking- 
stake,  the  proper  link  hooked  over  the  staple,  and  the  block 
brought  to  its  true  position  by  the  screw.  This  position  is 
shown  by  the  bubble  L attached  to  the  top  of  the  block.  If 
the  lever-arms  HK  and  HK'  are  properly  proportioned,  the 
pull  on  the  tape  is  now  equal  to  the  weight  P.  To  find  this 
length  of  the  arm  HK,  let  HK  — k ; HK'  = k' ; the  horizontal 
distance  from  the  knife-edge  H to  the  centre  of  gravity  of  the 
block  —g\  and  the  weight  of  block  “ B. 

Then,  taking  moments  about  H,  we  have 

Pk  — Pk' Bg  or  k = k' g.  . . . . (i) 

When  equation  (i)  is  fulfilled  then  the  pull  on  the  tape  is  just 
equal  to  the  weight  P,  when  the  bubble  reads  horizontally.  The 
centre  of  gravity  of  the  block  is  found  by  suspending  it  from 
two  different  axes  and  noting  the  intersection  of  plumb-lines 
dropped  from  these  axes. 

At  the  rear  end  the  tape  is  held  by  a slide  operated  by  an 
adjusting  screw  similar  to  that  shown  in  Fig.  129.  This  slide 
rests  on  the  straining-stake,  and  the  rear-end  graduation  is 
made  to  coincide  exactly  with  the  graduation  on  the  zinc 
strip  which  marked  the  forward  end  of  the  previous  tape- 
length.  The  rear  observer  gives  the  word,  and  the  forward  end 
is  marked  on  the  next  zinc  strip.  The  thermometers  are  then 
read,  and  the  tape  carried  forward. 

The  measurement  is  duplicated  by  measuring  again  in  the 
same  direction,  the  zinc  strips  being  left  undisturbed. 

In  obtaining  a profile  of  the  line  the  level  rod  is  held  on 
the  suspension  nails  and  on  a block,  equal  in  height  to  the 
length  of  the  hooks,  set  on  top  of  the  marking-stakes. 


GEODETIC  SURVEYING, 


453 


For  transferring  the  work  to  the  ground,  or  to  a stone  set 
beneath  the  surface,  a transit  is  mounted  at  one  side  of  the 
line  and  the  point  transferred  by  means  of  the  vertical  motion 
of  the  telescope,  the  line  of  sight  being  at  right  angles  to  the 
base-line. 

340.  M.  Jaderin’s  Method. — Prof.  Edward  Jaderin,  of 
Stockholm,  has  brought  the  measurement  of  distances  by 
wires  and  steel  tapes  to  great  perfection.  He  uses  a tape  25 
metres  in  length,  and  stretches  it  over  tripods  set  in  line,  as 
shown  in  Fig.  130.  On  the  top  of  the  tripod  head  is  a fixed 
graduation.  At  the  rear  end  of  the  tape  there  is  a single  grad- 
uation, but  at  the  forward  end  a scale  ten  centimetres  in  length 


is  attached  to  the  tape,  this  being  graduated  to  millimetres  on 
a bevelled  edge.  The  middle  of  this  scale  is  25  metres  from  the 
graduation  at  the  other  end  of  the  tape.  The  tripods  are  set 
as  near  as  may  be  to  an  interval  of  25  metres,  but  it  is  evident 
that  the  reading  may  be  taken  on  them  if  this  interval  is  not 
more  than  5 centimetres  more  or  less  than  25  metres.  The 
reading  is  taken  to  tenths  of  millimetres,  the  tenths  being 
estimated.  The  tape  is  stretched  by  two  spring  balances,  a 
very  stiff  spring  being  used  at  the  rear  end  and  a very  sensi- 
tive one  at  the  forward  end.  The  rear  balance  simply  tells  the 
operator  here  when  the  tension  is  approximately  right,  the 
measure  of  this  tension  being  taken  on  the  forward  balance, 
which  is  shown  in  the  figure. 


454 


SURVEYING. 


If  a single  steel  wire  or  tape  be  used,  Mr.  Jadcrin  also 
finds  that  the  work  must  be  done  in  cloudy  and  calm  weather, 
or  at  night,  if  the  best  results  arc  to  be  obtained.  But  he 
finds  that  if  two  wires  be  used,  one  of  steel  and  the  other  of 
brass,  he  can  continue  the  work  during  the  entire  day,  even  in 
sunshine  and  wind,  and  obtain  an  accuracy  of  about  one  in 
one  million  in  his  results.*  The  wires  are  stretched  in  succes- 
sion over  the  same  tripods,  by  the  same  apparatus,  one  wire 
resting  on  the  ground  while  the  other  is  stretched.  More  ac- 
curate results  could  doubtless  be  obtained  if  both  wires  are 
kept  off  the  ground  constantly,  the  wire  not  in  use  being  held 
by  two  assistants,  or  if  stakes  and  wire  hooks  are  used,  both 
wires  might  be  stretched  at  once  in  the  same  hooks.  The  two 
wires  form  a metallic  thermometer,  the  difference  between  the 
readings  of  the  same  distance  by  the  two  wires  determining 
the  temperature  of  both  wires,  when  their  relative  lengths  at  a 
certain  temperature  and  their  coefficients  of  expansion  are 
known.  This  method  is  similar  in  principle  to  that  of  the 
Coast  Survey  apparatus,  where  steel  and  zinc  bars  are  used, 
shown  in  Fig.  128.  In  such  cases  the  true  length  of  line  is 
found  by  equation  (5),  p.  461. 

At  least  three  thermometers  should  be  used  on  a 300-foot 
tape,  and  they  should  be  lashed  to  the  tape  or  suspended  by  it  at 
such  points  as  to  have  equal  weight  on  determining  its  tempera- 
ture. Thus  if  the  tape  is  300  feet  long  the  thermometers  should 
be  fastened  at  the  50,  150,  and  250  foot  marks.  They  should  of 
course  have  their  corrections  determined  by  comparison  with 
some  absolute  standard  or  with  other  standardized  thermom- 
eters. 


* See  “ Geodatische  Langenmessung  mit  Stahlbanden  and  metalldrahien,” 
von  Edv.  Jaderin,  Stockholm.  1885.  57  pp.  Also,  “ Expose  el^mentaire  de 
la  nouvelle  Methode  de  M.  Edouard  Jaderin  pour  la  mesure  des  droites  ge'ode- 
siques  au  moyen  de  Bandes  d’Acier  et  de  Fils  metalliques,”  par  P.  E.  Bergstrand, 
Ingenieur  au  Bureau  central  d’Arpentage,  i Stockholm.  1885.  48  pp. 


GEODETIC  SURVEYING. 


455 


If  the  appliances  above  outlined  be  used  with  a single  tape 
or  wire,  and  the  work  be  done  on  calm  and  densely  cloudy 
days,  or  at  night,  or  with  two  wires  used  even  in  clear  weather, 
it  is  not  difficult  to  make  the  successive  measurements  agree  to 
an  accuracy  of  one  in  five  hundred  thousand.  There  still  re- 
mains, however,  the  errors  in  the  absolute  length,  in  the  coeffi- 
cient of  expansion,  in  the  modulus  of  elasticity,  in  the  measure 
of  the  pull,  and  in  the  alignment,  none  of  which  would  appear 
in  the  discrepancies  between  the  successive  measurements. 

341.  The  Absolute  Length  is  the  most  difficult  to  deter- 
mine. The  best  way  of  finding  it  would  be  to  compare  it  with 
another  tape  of  known  length.  This  may  be  impracticable, 
since  all  the  so-called  “ standards”  are  more  or  less  discrepant 
on  coming  from  the  makers.* 

If  an  absolute  standard  is  not  available,  then  the  length  may 
be  found  by  measuring  a known  distance,  as  a previously 
measured  base-line,  and  computing  the  temperature  at  which 
the  tape  is  standard.  Or  the  tape  may  be  compared  with  a 
shorter  standard,  as  a yard  or  metre  bar,  by  means  of  a com- 
parator furnished  with  micrometer  microscopes.f 

* The  absolute  length  of  the  300-foot  steel  tape  belonging  to  the  Mississippi 
River  Commission,  the  coefficient  of  expansion  and  the  modulus  of  elasticity  of 
which  the  author  himself  determined  in  1880,  has  now  been  obtained.  This  was 
done  by  measuring  a part  of  the  Onley  Base  Line  with  this  tape,  using  the 
method  herein  outlined.  This  base  is  situated  in  Southern  Illinois,  and  forms 
the  southern  extremity  of  U.  S.  Lake  Survey  primary  triangulation-system.  The 
probable  error  in  the  length  of  the  base,  from  the  original  measurements,  was 
about  one  one-millionth.  The  recent  tape-measurements  are  remarkably  accor- 
dant, so  the  length  of  this  tape  is  now  very  accurately  known.  A similar  tape 
belonging  to  the  engineering  outfit  of  Washington  University  has  been  com- 
pared with  this  one  at  different  temperatures,  and  its  absolute  length  and  coeffi- 
cient of  expansion  found.  The  50-foot  subdivisions  have  also  been  carefully 
determined. 

f Such  an  apparatus  is  used  in  the  physical  laboratory  of  Washington  Uni- 
versity, which,  in  conjunction  with  a standard  metre  bar  which  has  been  com- 
pared with  the  European  standards,  enables  absolute  lengths  to  be  determined 
to  the  nearest  one-thousandth  of  a millimetre. 


456 


SUR  VE  YING. 


342.  The  Coefficient  of  Expansion  may  be  taken  any- 
where from  0.0000055  to  0.0000070  for  1°  F.*  If  the  tape  is 
used  at  nearly  its  standard  temperature,  then  the  coefficient  of 
expansion  plays  so  small  a part  that  its  exact  value  is  unim- 
portant. If  it  is  used  at  a temperature  of  70°  F.  from  its 
standard  temperature,  and  if  the  error  in  the  coefficient  used 
be  twenty  per  cent,  the  resulting  error  in  the  work  would  be 
one  in  ten  thousand.  This  is  probably  the  extreme  error  that 
would  ever  be  made  from  not  knowing  the  coefficient  of  ex- 
pansion, some  tabular  value  being  used.  If  nothing  is  known 
of  the  coefficient  of  expansion,  probably  o.cmX)Oo65  would  be 
the  best  value  to  use.  It  is  evident,  however,  that  for  the 
most  accurate  work  the  coefficient  of  expansion  of  the  tape 
used  must  be  carefully  determined. 

* The  author  made  a series  of  observations  on  a steel  tape  300  feet  long,  the 
readings  being  taken  at  short  intervals  for  four  days  and  three  nights.  The 
tape  was  enclosed  in  a wooden  box,  and  supported  by  hooks  every  sixteen 
feet.  The  observations  were  taken  on  fine  graduations  made  by  a diamond 
point,  there  being  a single  graduation  at  one  end,  but  some  fifty  graduations  a 
millimetre  apart  at  the  other  end.  The  readings  were  made  by  means  of 
micrometer  microscopes  mounted  on  solid  posts  at  the  two  ends.  The  range 
of  temperature  was  about  50°  F.,  and  the  resulting  coefficient  of  expansion  for 
1°  F.  was  0.00000699  ± 3 in  the  last  place.  The  coefficient  for  the  Washington 
University  tape  is  0.00000685.  Prof.  T.  C.  Mendenhall  found  from  six  or  eight 
experiments  on  steel  bands  used  for  tapes,  a mean  coefficient  of  0.0000059. 
Steel  standards  of  length  have  coefficients  ranging  from  0.0000048  to  0.0000066. 

Mr.  Edward  Jaderin,  Stockholm,  has  obtained  a mean  value  of  0.0000055, 
from  a number  of  very  careful  determinations,  both  from  remeasuring  a primary 
base-line,  and  from  readings  in  a water-bath.  Several  steel  wires  were  tested, 
and  their  coefficients  all  came  very  near  the  mean  as  given  above. 

For  brass  zvires  he  found  a mean  coefficient  of  0.0000096  F.  The  15-foot 
standard  brass  bar  of  the  U.  S.  Lake  Survey  has  a coefficient  of  0.0000100, 
while  tabular  values  are  found  as  high  as  0.0000107  F. 

There  is  some  evidence  that  cold-drawn  wires  have  a less  coefficient  of  expan- 
sion than  rolled  bars  and  tapes. 

Coefficients  of  expansion  have  seldom  been  found  with  great  accuracy,  the 
coefficients  of  the  “ M6tre  des  Archives,”  the  French  standard,  having  had  an 
erroneous  value  assigned  to  it  for  ninety  years 


GEODETIC  SURVEYING. 


457 


343.  The  Modulus  of  Elasticity  is  readily  found  by  ap- 
plying to  the  tape  varying  weights,  or  pulls,  and  observing  the 
stretch.  The  correction  for  sag  will  have  to  be  applied  for  each 
weight  used,  in  case  the  tape  is  suspended  from  hooks,  which 
should  be  done  to  eliminate  all  friction. 

Let  be  the  maximum  load  in  pounds ; 

Pq  “ “ minimum  load  in  pounds  ; 
a “ “ increased  length  of  tape  in  inches  due  to  the 
increased  pull ; 

L “ “ length  in  inches  for  pull  P^,  or  the  graduated 

length  of  tape ; 

S “ “ cross-section  in  square  inches ; 

E ‘‘  “ modulus  of  elasticity ; 

d “ “ distance  between  supports ; 

w “ “ weight  of  one  inch  of  tape  in  pounds  ; 

s “ “ shortening  effect  of  the  sag  for  the  length  L ; 

V “ “ sag  in  inches  midway  between  supports. 


Then  we  have 


(^.  - p:)L 

Sa 


(I) 


But  for  the  pull  Pj,  the  shortening  from  sag  is  much  less 
than  for  the  pull  P^.  We  must  therefore  find  the  effect  of  the 
sag  in  terms  of  the  pull. 

344.  Effect  of  the  Sag. — Where  the  sag  is  small,  as  it 
always  is  in  this  work,  the  curve,  although  a catenary,  may  be 
considered  a parabola  without  an  appreciable  error. 

If  we  pass  a section  through  the  tape  midway  between  sup- 
ports, and  equate  the  moments  of  the  external  forces  on  one 
side  of  this  section,  we  obtain,  taking  centre  of  moments  at 
the  support, 

wd  d wd'' 


or 


Pv  = — .--  ^ 
24  8 


zvd'" 


ZP' 


(0 


458 


SURVEYING. 


$ 


If  the  length  of  a parabolic  curve  be  given  by  an  infinite 
series,  and  if  all  terms  after  the  second  be  omitted,  which  they 

may  when  ^ is  small,  then  we  may  write — 

Length  of  curve  :=  d 


If  we  now  substitute  for  v its  value  as  given  in  equation 
(i),  we  have 


Length  of  curve  = ^ | ^ 


If  we  call  the  excess  in  length  of  curve  over  the  linear  dis- 
^ tance  between  supports  the  effect  of  the  sag.,  we  have 


d I wdV 

24\PJ 


(3) 


for  one  interval  between  supports.  If  there  are  n such  inter- 
vals in  one  tape-length,  then  7id  = Z,  and  the  effect  of  the  sag 
in  the  entire  tape-length  is 


(4) 


If  5,  and  be  the  effects  of  the  sag  for  the  pulls  Z,  and 
(5i<5o  forP,>Po),  then  the  total  movement  at  the  free  end 
due  to  the  pull  being  increased  from  P^  to  P,  would  he  a -\- 
(S,  — 5,).  If  this  total  movement  be  called  M,  then  we  would 
have 


£ = 


S(M-\  + S.) 


P. 


m {wdf 

\L~  24 

V p^p:  j) 

(5) 


GEODETIC  SURVEYING. 


459 


Example. 


Let  P\  = 6o  pounds; 

/’o  = 10  pounds; 

^i}  = 0.00055  pound  per  inch  of  tape; 
d — 300  inches  = 25  feet; 

S — 0.002  square  inch; 

M — 3.2  inches; 

Z = 3600  inches  = 300  feet. 

To  find  E. 

From  equation  (5)  we  have 


E = 


— — 28,500,000. 


3.2  0.027/  3500 

3600  24  UfioocK 


0.002 


From  the  same  data,  we  find  from  eq.  (4)  the  effect  of  the  sag  to  be  0.040 
inch  for  the  ten-pound  pull,  and  0.001  inch  for  the  sixty-pound  pull. 

Evidently,  if  the  tape  is  stretched  by  the  same  weight  when  its  absolute 
length  is  found,  and  when  used  in  measuring,  the  stretch,  or  elongation  from 
pull,  would  not  enter  in  the  computation,  and  so  the  modulus  of  elasticity 
would  be  no  function  of  the  problem. 

Again,  the  stretch  per  pound  of  pull  may  be  observed  for  the  given  tape,  and 
then  neither  E nor  S,  the  cross-section,  would  enter  in  the  computation. 

345.  Temperature  Correction. — If  mercurial  thermome- 
ters are  used,  their  field-readings  must  first  be  corrected  for 
the  errors  of  their  scale-reading,  each  thermometer  having,  of 
course,  a separate  set  of  corrections.  Then  the  mean  of  the 
corrected  readings  may  be  taken  for  all  the  whole  tape-lengths 
in  the  line  measured,  and  the  correction  for  the  entire  line 
obtained  at  once.  Thus, 

let  L = length  of  line  ; 

7*0  = temperature  at  which  the  length  of  the  tape  is  given 
for  the  standard  pull  this  usually  being  the  tem- 
perature at  which  its  true  length  is  its  graduated 
length  for  that  standard  pull ; 

Tm  = the  mean  corrected  temperature  of  the  entire  line ; 
a = coefficient  of  expansion  for  1°  ; 

Q = correction  for  temperature. 


460 


SURVEYING. 


Then  C,^^a[T^-T:)L (i) 

The  temperature  correction  for  a part  of  a tape-length  is  com- 
puted separately. 

If  the  value  of  01  for  the  tape  used  is  not  known,  it  may  be 
taken  at  0.0000065. 

If  a metallic  thermometer  is  used,  as  a brass  and  a steel 
wire,  or  a brass  and  a steel  bar  as  in  the  U.  S.  C.  and  G.  S. 
apparatus  shown  on  p.  446,  then  we  have  the  following: 

346.  Temperature  Correction  when  a Metallic  Ther- 
mometer is  used. 


Let  / = length  of  wire  or  tape  used,  as  300  feet ; 

4 = absolute  length  of  the  steel  wire  at  the  standard 
temperature  of,  say,  32°  F. ; 

4 = same  for  brass  wire  ; 

L — total  length  of  line  for  whole  tape-lengths  ( = nl 
approximately)  ; 

n — number  of  lengths  of  the  standard  measured ; 

Tg  — mean  value  of  all  the  scale-readings  on  steel  wire 


for  the  entire  line 


= same  for  scale-readings  on  brass  wire ; 
as  = coefficient  of  expansion  for  the  steel  wire 
a^  — “ “ “ “ brass  “ 

4 = mean  temperature  for  the  entire  line. 


Then  we  have 


Z = K4  + r,)(i+(/,- 32°K)) 

= »(4  + ^,)(i+(;.-32>,))  • • • • W 

Since  the  temperature  correction  is  relatively  a very  small 
quantity,  we  may  put  4 -|-  = 4 + ^6  = 4 the  length  of  the 

tape  to  which  the  temperature  correction  is  applied. 


GEODETIC  SURVEYING. 


461 


We  then  have  from  (2) 


{t.  - 32“)  = 


(4  "h  ^s)  ~ (4  ~f~ 


. . (3) 


Substituting  this  value  of  the  temperature  in  (2),  we  obtain 

L — n[ls  + ^ ((4  + ^s)  — (4  + ^b))]-  • (4) 

If  we  put  4 -f  = 5s  and  4 + ^&  = ‘S'5,  we  have 

£ = .[i.  + (5,-5.)5^J' 

• • • 


(5) 


From  either  of  the  equations  (5)  we  may  compute  the  length 
of  the  line  as  corrected  for  temperature.  If,  however,  it  is 
desired  to  find  the  temperature  correction  separately,  in  order 
to  combine  it  with  the  other  corrections,  we  have 


— ^(‘5's  55)  , .....  (6) 

— ^3 

for  the  temperature  correction  to  be  applied  to  the  measured 
length  by  the  steel  wire,  or 


(7) 


as  the  temperature  correction  to  be  applied  to  the  measured 
length  by  the  brass  wire. 


462 


SUI^  VE  YING. 


These  formulae  all  apply  only  to  the  entire  tape-lengths.  Any 
fractional  length  would  have  to  be  computed  separately,  or 
else  a diminished  weight  given  to  their  scale-readings  in  obtain- 
ing the  mean  values,  r,  and 

347.  Correction  for  Alignment,  both  horizontal  and  ver- 
tical.— The  relative  elevations  of  the  points  of  support  are 
found  by  a levelling  instrument,  and  the  horizontal  alignment 
done  by  a transit  or  by  eye.  An  alignment  by  eye  will  be 
found  sufficiently  exact  if  points  be  established  on  line  by 
transit  every  500  or  locx)  feet.  The  suspending  nails  and  hooks 
afford  considerable  latitude  for  lateral  adjustment  when  the 
tape  is  stretched  taut ; hence  the  horizontal  deviation  will  be 
practically  zero  unless  the  stakes  are  very  badly  set,  and  the 
relative  elevations  of  any  two  successive  supports  should  be 
determined  to  less  than  0.05  foot.  If  no  care  is  taken  to  have 
more  than  two  suspension  points  on  grade,  then  each  section 
of  the  tape  will  have  a separate  correction.  Usually  a single 
grade  may  as  well  extend  over  several  sections,  in  which  case  the 
portion  on  a uniform  grade  may  be  reduced  as  a single  section. 

Let  /,,  4,  4,  etc.,  be  the.  successive  lengths  of  uniform  grades, 
and  h^y  h^y  etc.,  the  differences  of  elevation  between  the 
extremities  of  these  uniform  grades ; then  for  a single  grade  we 
would  have  the  correction 


C = / _ IP 

p _ 2CI+C  -h\ 


or 


But  since  is  a very  small  quantity  as  compared  with  4 

jp 

we  may  drop  the  O,  whence  we  have  C = for  a single  grade. 
The  exact  value  of  Cy  in  ascending  powers  of  hy  is 


(0 


GEODETIC  SURVEYING. 


463 


For  the  entire  line,  if  all  but  the  first  term  be  neglected, 
the  correction  is 


If  the  /’s  are  all  equal,  as  when  no  two  successive  suspen- 
sion points  fall  in  the  same  grade,  then  we  have 


G — “ + ^^3"  + • • • • (3) 


Since  the  relative  elevations  are  determined,  and  not  the 
angles  of  the  grades,  these  formulae  are  more  readily  applied 
than  one  involving  the  grade  angles. 

The  error  made  in  rejecting  the  second  power  of  C in  the 
above  equations  is  given  in  the  table  on  the  following  page, 
where  / and  h are  taken  in  the  same  unit  of  length.* 

If  the  grades  are  given  in  vertical  angles,  as  they  always 
are  with  the  ordinary  base  apparatus,  then  we  have  for  the 
correction  to  each  section  whose  length  is  /,  and  whose  grade 
is  6 above  or  below  the  horizon. 


Cg  = —l{i  — COS  6) 


2/  sin"  -. 
2 


If  6 be  expressed  in  minutes  of  arc,  and  if  the  grade  angle 
is  less  than  about  six  degrees,  or  if  the  slope  is  less  than  one  in 
ten,  we  may  write  • 


sin  I 


2 


= —0.00000004231  OH; 


* From  Jaderin’s  Geodiitische  Liingenmessung. 


464 


SUR  VE  YING. 


or  by  logarithms, 

log  Cg  — const,  log  2.626422  + 2 log  0 log  /. 


TABLE  OF  RELATIVE  ERRORS  IN  THE  FORMULA  Cg=z 

2/ 


Length  of 

Relative  Error  = 

Grade. 

1 1 

1 

0.00005 

0.00015 

0.00025 

0.00035 

1 0.00045 

/. 

h = Rise  or  Fall  in  Length  /. 

I 

0.14 

0. 19 

1 

2 

.24 

.31 

1 

“2 

.32 

.42 

i 

. .40 

• 53 

•47 

.62 



D 

6 

. 54 

.71 

0.81 

7 

.61 

.80 

.91 

i 

8 

.67 

•88 

1 .00 

9 

•73 

.97 

1 . 10 

1. 19 

10 

•79 

1.05 

1. 19 

1.29 

II 

• 85 

1 . 12 

1.28 

1-39 

12 

.91 

1.20 

1.36 

1 .48 

13 

•97 

1.27 

1-45 

1-57 

j 1.67 

14 

1.02 

1-35 

1-53 

1.66 

1-77 

15 

1.08 

1.42 

1 .61 

1-75 

1.86 

16 

1. 13 

1.49 

1.69 

1.84 

1.96 

17 

1. 18 

1.56 

1-77 

1.92  i 

2.05 

18 

1.24 

1.62 

1.85 

2.01 

2.14 

19 

1.29 

1.69 

1.92  1 

2.09 

2.23 

20 

1-34 

1.76 

2.00  i 

2.17 

2-31 

21 

1-39 

1.82 

2.07 

2.25 

2.40 

22 

1.44 

1.89 

2.15 

2-33 

2.48 

23 

1.48 

1-95 

2.22 

2.41 

2-57 

24 

1-53 

2.02 

2.29 

2.49 

2.65 

25 

1.58 

2.08 

2.36 

2.57 

2-73 

26 

1.63 

2.14 

2.43 

2.65 

2.82 

27 

1.67 

2.20 

2.50 

2.72 

2.90 

28 

1.72 

2.26 

2-57 

2.80 

2.98 

29 

1.77 

2.32 

2.64 

2.87 

3 -06 

30 

1. 81 

2.38 

2.71 

2.95 

3-14 

GEODETIC  SURVEYING. 


465 


( 

348.  Correction  for  Sag.— From  equation  (4),  p.  458,  we 
have 


L fwd'^ 


(4) 


If  the  standard  length  be  given  with  the  pull  and  the 
distance  between  supports  while  in  the  field  the  pull  P and 
distance  d between  supports  be  used,  then  the  correction  for 
sag  is 


LzP  fd," 

24  \p: 


L_ 

24 


. (5) 


where  Z,  d,  and  are  taken  in  the  same  unit  of  length,  and  w 
is  the  weight  of  a unit’s  length  of  tape  in  the  same  units  used 
forP. 

349.  Correction  for  Pull.— From  equation  (i),  p.  457,  we 
may  write  at  once 


, {p-p:)L 

SE  ■ 

Here  P is  taken  in  pounds,  Z and  Cp  in  inches,  and  S in 
square  inches,  since  E is  usually  given  in  inch-pound  units.  If 
E has  not  been  determined  by  experiment,  it  may  be  taken  at 
28000000.  The  cross-section  .S  is  best  found  by  weighing  the 
tape  and  computing  its  volume,  counting  3.6  cubic  inches  to 
the  pound.  Knowing  the  length,  the  cross-section  can  then  be 
found.  If  the  stretch  has  been  observed  for  different  weights, 
and  the  value  of  E computed,  the  value  of  .S  is  of  no  conse- 
quence, provided  the  same  value  be  used  for  both  observations. 

350.  Elimination  of  Corrections  for  Sag  and  Pull. — 
Since  the  correction  for  sag  is  negative  and  that  for  pull  is 
positive,  we  may  make  them  numerically  equal,  and  so  elimi- 
30 


466 


SUR  VE  YING. 


nate  them  both  from  the  work.  If  this  be  done,  the  normal 
or  standard  length  of  the  tape  should  be  obtained  for  no  sag 
and  no  pull,  and  its  normal  or  standard  temperature  found  such 
that  at  this  temperature,  and  for  no  sag  and  no  pull,  its  gradu- 
ated length  is  its  true  length. 

If  Z’o  is  the  temperature  at  which  the  tape  is  of  standard 
length  for  the  pull  and  the  distance  between  supports, 
and  if  I is  the  length  of  the  tape,  then  we  have. 


Shortening  from  sag  = - l-r,-)  , 

24  ' / 


Lengthening  from  x degrees  F.  ==  xaL 

If,  therefore,  the  effects  of  sag  and  pull  were  eliminated, 
the  tape  would  be  of  standard  length  at  a temperature 
degrees  above  where 


• • (I) 


where  all  dimensions  are  in  inches  and  weights  in  pounds. 

The  standard  temperature  for  no  sag  and  no  pull  would  be, 
therefore. 


7;  = 7;  + 


(2) 


We  will  call  this  the  normal  temperature. 


GEODETIC  SURVEYING. 


467 


In  order  that  the  corrections  for  sag  and  pull  shall  balance 
each  other,  we  must  have 


or 


EL 


which  we  will  call  the  normal  tension. 

If  the  stretch  in  inches  is  known  for  one  pound  of  pull  for 
the  given  tape,  we  may  call  this  and  we  will  have 

e — or  SE  = 

SE  e 


Also,  Iw  — W — weight  of  entire  tape  between  end  graduations, 
W 

or  w = 

I 

And  -^=71  = number  of  sags  in  the  tape. 

Substituting  these  values  in  (3),  we  obtain 


where  W = weight  of  entire  tape  in  pounds ; 

I = length  of  tape  in  inches  ; 
e = elongation  of  tape  for  a one-pound  pull ; 

I 

n = number  of  sags  in  tape  = 

If  the  tape  has  no  intermediate  supports,  then  n—i,  and 
we  have  for  the  normal  tension 


(5) 


468 


scr/c  VE  YING. 


Example. — For  the  300-foot  steel  tape,  whose  constants  the  author  deter- 
mined, we  have  IV  = 2 lbs.,  / = 3600  inches,  e = 0.066  inch.  If  the  supports 
are  30  feet  apart,  n = 10,  whence,  from  eq.  (4),  Pn  = 4.48  pounds. 

If  n = 6,  or  if  the  supports  were  placed  50  feet  apart,  we  would  find  I'n  = 
6.32  pounds. 

If  n = 3,  or  if  the  supports  are  too  feet  apart,  Pn  = 10.03  pounds. 

In  the  last  case,  the  sag  would  be  ten  inches  midway  between  supports. 


351.  To  reduce  a Broken  Base  to  a Straight  Line.— 
It  is  sometimes  nece.ssary  or  convenient  to  introduce  one  or 
more  angles  into  a base-line.  These  would  never  deviate  much 
from  180°.  Let  the  difference  between  the  angle  and  180°  be 
6,  and  let  the  two  measured  sides  be  a and  b,  to  find  the  side  c. 
If  6 be  expressed  in  minutes  of  arc  and  if  it  is  not  more  than 
about  3°,  the  following  approximate  formula  will  prove  suf- 
ficiently exact : 


side  c = 


a-\-  b — 


ab&^ 

a-\-b 


a b — 0.00000004231 


abe^ 

a-\-b' 


If  S is  greater  than  from  3°  to  5°,  the  triangle  would  have  to 
be  computed  by  the  ordinary  sine  formula. 

352.  To  reduce  the  Length  of  the  Base  to  Sea-level. 
— In  geodetic  work,  all  distances  are  reduced  to  what  they 
would  be  if  the  same  lines  were  projected  upon  a sea-level 
surface  by  radii  passing  through  the  extremities  of  the  lines. 
It  is  not  necessary,  however,  to  reduce  all  the  lines  of  a trian- 
gulation system  in  this  manner,  since  if  the  length  of  the  base- 
line is  so  reduced  the  computed  lengths  of  all  the  other  lines 
of  the  system  will  be  their  lengths  at  sea-level.  The  angles 
that  are  measured  are  the  horizontal  angles,  and  are  not  affected 
by  the  differences  of  elevation  of  the  various  stations.  It  is 


GEODETIC  SURVEYING. 


469 


necessary,  therefore,  to  know  the  approximate  elevation  of  the 
base  above  sea-level. 

Let  r = mean  radius  of  earth ; 

a =:  elevation  above  sea-level ; 

B — length  of  measured  base ; 
b — length  of  base  at  sea-level. 

Then  r a \ r ::  B : b, 


or 


b = B 


r 

r-\-  a 


The  correction  to  the  measured  length  is  always  negative, 
and  is 


C=b-B= - B 


B 


^ \ 


Since  a is  very  small  as  compared  to  r,  we  may  write 


The  mean  radius*  in  feet  is 


mean  r = 


20926062  + 20855121 
2 


= 20890592  feet, 


log  r (in  feet)  = 7*3199507. 


353.  Summary  of  Corrections. — For  the  significance  of 
the  notation  used  in  the  following  equations,  see  the  preceding 
articles  where  they  are  derived.  The  corrections  are  all  for 


* Rigidly,  we  should  use  ihe  length  of  the  normal  for  the  given  latitude,  but 
the  mean  radius  as  above  found  is  sufficient  for  most  cases. 


4/0 


SUK  VE  Y I NG. 


the  entire  line  measured,  or  rather  for  that  portion  of  it  com- 
posed of  entire  tape-lengths,  and  are  to  be  applied  with  the 
signs  given  to  the  measured  length. 

I.  Correction  for  Temperature. 

For  a single  standard  with  mercurial  temperatures, 

C,  = + «(7'„-  7;)Z (I) 

For  metallic  thermometer-readings,  as  found  from  steel  and 
brass  standards,  for  instance,  the  correction  to  be  applied  to 
the  length  as  found  by  the  steel  wire,  or  standard,  is 


= (2) 

2.  Correction  for  Grade. 

In  terms  of  the  difference  of  elevation  of  grade,  points  at  a 
common  distance,  /,  apart, 


= (3) 

In  terms  of  the  grade  angles,  expressed  in  minutes  of  arc, 

Cg  = — o.oooocxx)423i.2’6'V. (4) 

3.  Correction  for  Sag. 

For  the  standard  length  given  for  a pull  and  a distance 
between  supports  while  P and  d are  used  in  the  field-work, 


(5) 


GEODETIC  SURVEYING. 


471 


For  the  standard  length  given  for  no  pull  and  no  sag, 


L (wdV 
24  \P  1 ’ 


(6) 


4.  Correction  for  Pull. 


P-P. 

SE 


(7) 


or  C^—{P—  Po)en. 


(8) 


5.  To  reduce  Standard  Temperature  to  Normal  Temperature. 

When  the  temperature  of  the  tape  {T^  is  known  at  which 
the  graduated  is  the  true  length  for  the  pull  P^  and  distance 
between  supports  to  find  the  corresponding  temperature  for 
no  pull  and  no  sag,  this  being  called  the  nor^nal  temperature 
{Tn),  we  have,  in  degrees. 


_!  Izudiy- 

2i\PjJ 


• • (9) 


6.  To  eliminate  Correctioyis  for  Sag  and  Pull. 


or 


Make  the  pull  P^  = 


Pn=^ 


(10) 


00 


For  no  intermediate  supports  to  tape. 


472 


SUA^  VE  YING. 


is  called  the  normal  tc7isio7i. 

7.  Correction  for  Broke7i  Base. 

If  a and  b are  the  two  measured  sides  which  make  an  angle 
of  180°  — the  correction  to  be  added  to  a b to  get  the 
distance  between  their  extremities,  6 being  less  than  5'’,  and 
expressed  in  minutes  of  arc,  is 


abO^ 


a 


8.  Correctio7i  to  Sea-level. 


where  L is  the  length  of  the  measured  base  at  an  altitude  a 
above  sea-level. 


log  r (in  feet)  = 7*3199507. 


354.  To  compute  any  Portion  of  a Straight  Base  which 
cannot  be  directly  measured. — It  sometimes  is  convenient 


A 


x> 


o 


Fig.  131. 


to  take  a base-line  across  a stream  or  other  obstruction  to  di- 
rect measurement.  In  such  a case  a station  may  be  chosen 


GEODETIC  SURVEYING. 


473 


as  O in  Fig.  131,  and  the  horizontal  angles  A OB  = P,  BOC  = 
Q,  and  COD  = R measured.  If  the  parts  AB  and  CD  lie  in 
the  same  straight  line,  and  AB  = a and  CD  — b are  known, 
then  BC  = x may  be  found  by  measuring  only  the  angles  at  O. 

Thus  in  the  triangles  ABO  and  ACO  we  have 

CO  _ X -^a  sin  P 
m ~ sin  (P+"0’ 


also  from  the  triangles  BDO  and  CDO  we  have 

CO  _ b sin  {Q  + R) 

BO  ~ X b sin  P 

Let  K = P-\-  Q and  L—  Q + P,  then  by  equating  the 
above  values  of  -SS  we  have 


whence 


{x+a){x  + b)  = 


ab  (sin  K sin  L) 
sin  P sin  P ^ 


a-\-b  jabism  K sin  L)  , fa  — bV 

^ = — r-  ±V  + {-^1  • 

Evidently  only  the  positive  result  is  to  be  taken. 

The  points  A,  O,  and  D should  be  chosen  so  as  to  give 
good  intersections  at  A and  D. 

355.  Accuracy  attainable  by  Steel-tape  and  Metallic- 
wire  Measurements. — The  following  results  have  been  at- 
tained by  using  the  methods  herein  described : 

I.  In  Sweden,  Mr.  Edw.  Jaderin  measured  a primary 
base-line  two  kilometres  in  length  three  times,  by  means  of 
steel  and  brass  wires  25  metres  long,  in  ordinary  summer 


474 


SUJ^VEV/NG. 


weather,  mostly  clear,  with  a probable  error  of  a single  deter- 
mination of  I in  600,000,  and  a probable  error  of  the  mean  re- 
sult of  I in  1,000,000,  as  compared  with  the  true  length  of  the 
line  as  obtained  by  a regular  primary  base  apparatus.* 

2.  On  the  trigonometrical  survey  of  the  Missouri  River, 
in  1885,  O.  B.  Wheeler,  U.  S.  Asst.  Engineer,  obtained 
the  following  results,  using  one  steel-tape  300  feet  long: 


First  measurement. 
Second  “ 

Glasgow  Base. 

7923-237  feet. 

7923-403  “ 

Mean 

In  this  case  the  sun  was  shining  more  or  less  on  both 
measurements.  The  probable  error  of  a single  result  is  i in 
100,000,  and  of  the  mean  of  two  measurements  i in  140,000. 


First  measurement. 
Second  “ 

Benton  Base. 

9870-443  feet. 

Mean 

The  probable  error  of  a single  measurement  is  i in  380,000, 
and  of  the  mean,  i in  533,000. 


First  measurement. 
Second  “ 

Trovers  Point  Base. 

9711.892  “ 

Mean 

971 1.904  ± 0.0078  feet. 

* For  title  of  Mr.  Jaderin’s  pamphlet  describing  his  methods  and  results,  see 
foot-note,  p.  454. 


GEODETIC  SURVEYING. 


475 


The  probable  error  of  a single  measurement  is  i in  900,000, 
and  of  the  mean  it  is  i in  1,250,000. 

Olney  Base. 

First  measurement 10821.9658  feet. 

Second  “ 10821.9665  “ 

Mean 10821.9662  ± 0.0002  feet. 

This  base  had  been  measured  by  the  U.  S.  Lake  Survey 
Repsold  base  apparatus,  with  a probable  error  of  about  i in 
1,000,000.  This  portion  of  it,  about  half  the  entire  base,  was 
remeasured  with  the  tape  in  order  to  determine  the  absolute 
length  of  the  tape.  The  work  was  done  on  both  the  tape- 
measurements  in  a drizzling  rain,  so  that  the  temperatures 
were  obtained  with  great  accuracy.  The  mean  tempera- 
tures of  the  two  measurements  differed,  however,  by  several 
degrees,  so  that  the  two  sets  of  graduations  on  the  zinc  strips 
were  quite  divergent,  and  it  was  only  after  the  final  reduc- 
tion that  the  two  results  were  known  to  be  so  nearly  identical.* 
3.  The  author  has  measured  a number  of  bases  about  one 
half  mile  in  length,  in  connection  with  students’  practice  sur- 
veys, by  the  methods  given  above,  and  in  each  case  obtained  a 
probable  error  of  the  mean  of  three  or  four  measurements  of 
less  than  one-millionth  part  of  the  length  of  the  line.  The 
work  was  always  done  on  densely  cloudy  days,  all  the  con- 
stants of  tape  and  thermometers  being  well  determined. 


* From  advance-sheets  of  the  Report  of  the  Missouri  River  Commission, 
1886. 


476 


SUR  VE  YING. 


Fig.  132. 


GEODETIC  SURVEYING. 


A77 


MEASUREMENT  OF  THE  ANGLES. 

356.  The  Instruments  used  in  triangulation  are  designed 
especially  for  the  accurate  measurement  of  horizontal  angles. 
This  demands  very  accurate  centring  and  fitting  at  the  axis,  and 
strict  uniformity  of  graduation.  It  was  formerly  supposed  that 
the  larger  the  circle  the  more  accurate  the  work  which  could 
be  done.  It  is  now  known  that  there  is  no  advantage  in  having 
the  horizontal  limb  more  than  ten  or  twelve  inches  in  diameter. 

There  are  two  general  methods  of  reading  fractional  parts 
of  the  angle,  smaller  than  the  smallest  graduated  space  on  the 
limb.  One  is  by  verniers,  the  other  by  micrometer  micro- 
scopes. Verniers  may  be  successfully  used  to  read  angles  to 
the  nearest  ten  or  twenty  seconds  of  arc,  but  if  a nearer  ap- 
proximation is  desired  microscopes  should  be  employed. 

Fig.  132  shows  a high  grade  of  vernier  transit,  capable  also 
of  reading  vertical  angles  to  70°.  Its  horizontal  limb  is  8 
inches  in  diameter  and  reads  by  verniers  to  ten  seconds.  It 
may  be  used  as  a repeating  instrument,  and  used  either  with 
or  without  a tripod.  To  mount  such  an  instrument  upon  a 
station  or  post,  a trivet,  made  of  brass  and  shown  in  Fig.  135,  is 
used.  The  pointed  steel  legs  are  driven  into  the  station,  the 
centre  of  the  opening  being  over  the  station  point.  The  arms 
have  angular  grooves  cut  in  their  upper  surface.  On  this  trivet 
may  be  set  any  three-legged  instrument,  so  long  as  the  radius 
of  its  base  is  not  greater  than  the  length  of  the  trivet  arms. 

In  Fig.  133  is  shown  a theodolite  (not  a transit  since  the 
telescope  does  not  revolve  on  its  horizontal  axis)  designed  for 
the  measurement  of  horizontal  angles  exclusively.  Here  mi- 
crometer microscopes  are  used.  The  horizontal  limb  is  from 
eight  to  twelve  inches  in  diameter.  There  is  no  vertical  circle 
or  arc,  so  that  no  vertical  angles  can  be  read.  Since  the  rela- 
tive heights  of  triangulation-stations  are  usually  determined 


* See  p,  484  for  explanation  of  this  term. 


478 


SUR  VE  YING. 


GEODETIC  SURVEYING. 


479 


I'iG.  134 


480 


SUR  VE  YING. 


from  their  relative  angular  elevations  or  depressions,  it  is  usual- 
ly necessary  to  have  a vertical  limb.  This  instrument  could 
not  be  used  as  a repeater. 

In  Fig.  134  is  shown  an  altazimuth  instrument,  or  an  in- 
strument designed  for  accurately  measuring  altitudes  as  well  as 
the  azimuths  of  points  or  lines.  Both  horizontal  and  vertical 
limbs  are  read  by  means  of  micrometer  microscopes.  Such  an 
instrument  is  designed  especially  for  astronomical  observations 
for  latitude  and  azimuth,  but  may  also  be  used  as  a meridian 

or  transit  instrument  for  observ- 
ing time  as  well  as  for  measuring 
horizontal  and  vertical  angles  in 
triangulation.  It  is  in  fact  the 
universal  geodetic  instrument, 
just  as  the  complete  engineer’s 
transit  is  the  universal  instrument 
in  ordinary  surveying.  In  almost 
all  cases  where  micrometers  are 
used  in  reading  the  angles  the 
limbs  are  graduated  to  five  or  ten  minutes  and  the  readings 
made  to  single  seconds. 

357.  The  Filar  Micrometer*  is  used  for  the  accurate  meas- 
urement of  small  distances  or  angles,  when  the  required  exact- 
ness is  greater  than  can  be  obtained  by  means  of  a vernier 
scale.  It  is  usually  combined  with  a microscope,  the  microme- 
ter threads  and  scale  lying  in  the  plane  of  the  image  produced 
by  the  objective.  This  image  is  always  larger  than  the  object 
itself  in  microscopes,  and  therefore  a given  movement  of  the 
wires  in  the  micrometer  corresponds  to  a ver}^  much  less  dis- 
tance on  the  object  sighted  at,  according  to  the  magnifying 
power  of  the  objective. 


* From  fdum,  thread;  micros,  small,  and  metros,  measure.  The  thread  is  in 
this  case  a spider’s  web,  or  scratches  on  glass. 


GEODETIC  SURVEYING. 


481 


The  frame  holding  the  movable  wires  has  a screw  with  a 
very  fine  thread  working  in  it,  called  the  micrometer  screw. 
This  screw  has  a graduated  cylindrical  head,  or  disk,  attached 
to  it,  there  usually  being  sixty  divisions  in  the  circumference 
when  used  in  angular  measurements.  The  number  of  whole 
revolutions  are  recorded  by  noting  how  many  teeth  of  a comb- 
scale  are  passed  over,  this  scale  being  nearly  in  the  plane  of  the 
wires  and  therefore  in  the  focus  of  the  eye-piece.  The  frac- 
tional parts  of  a revolution  are  read  on  the  graduated  screw- 
head  outside.  These  micrometer  attachments  are  shown  on 
the  two  microscopes  in  Fig.  133  and  on  the  five  in  Fig.  134. 


h 


Fig.  136. 


Fig.  136  is  a sectional  view  of  a filar  micrometer.  The  graduat- 
ed head  Ji  is  attached  to  the  milled  head  m,  forming  a nut  into 
which  the  micrometer-screw  a works.  This  screw  is  rigidly  at- 
tached to  the  frame  b,  to  which  are  fastened  the  movable  wires 
f . The  comb-scale  s and  fixed  wire  f are  attached  to  the 
frame  c,  which  is  adjusted  to  a zero-reading  of  the  graduated 
head  by  the  capstan-screw  d.  The  lost  motion  on  both  of 
these  frames  is  taken  up  by  springs.  The  complete  revolutions 
of  the  screw  are  counted  on  the  comb-scale,  and  the  fractional 
part  of  a revolution  on  the  graduated  head.  The  reading  is 
made  by  bringing  the  double  wires  symmetrically  over  a grad- 
uation, the  space  between  the  wires  being  a little  more  than 
the  width  of  the  graduation,  when  the  exact  number  of  revolu- 
tions and  sixtieths  are  read  on  the  comb-scale  and  on  the  head. 


31 


482 


SURVEYING. 


If  the  limb  is  graduated  to  ten  minutes  and  cacli  revolution 
corresponds  to  one  minute,  then  if  the  reading  is  taken  on  the 
nearest  graduation,  the  number  of  revolutions  need  never  ex- 
ceed five.  If,  however,  the  reading  be  always  taken  to  the  last 
ten-minute  mark  counted  on  the  limb,  then  ten  revolutions  may 
have  to  be  read  on  the  screw.  The  movement  of  the  threads 
is  as  they  appear  to  be,  tliere  being  no  inversion  of  image  be- 
tween wires  and  eye.  The  movement  on  the  limb  is,  however, 
opposite  from  the  apparent  motion. 

If  the  limb  is  graduated  to  ten  minutes,  and  a single  revo- 
lution of  the  screw  corresponds  to  the  space  of  one  minute, 
then  just  ten  revolutions  of  the  screw  should  move  the  wires 
from  one  graduation  to  the  next.  If  this  is  not  exactly  true, 
then  the  value  of  a ten-minute  space  should  be  measured  a 
number  of  times,  by  running  the  wires  back  and  forth,  the 
mean  result  taken,  and  from  this  the  value  of  one  revolution  of 
the  screw  determined.  This  value  is  called  the  “ run  of  the 
screw,”  and  a correction  is  applied  to  the  readings,  which  are 
always  made  in  degrees,  minutes,  and  seconds,  counting  one 
revolution  a minute  and  one  division  on  the  head  a second  of 
arc.  This  correction  is  called  “correction  for  run,”  and  should 
be  determined  for  all  parts  of  the  screw  used.  If  the  value  of 
one  revolution  is  not  exactly  what  it  is  designed  to  be,  it  can 
be  adjusted  by  moving  the  objective  of  the  microscope  in  or 
out  a little,  or  the  whole  microscope  up  or  down  with  refer- 
ence to  the  limb,  thereby  changing  the  size  of  the  image. 
Even  when  this  adjustment  is  accurately  made,  there  may  be 
still  a correction  for  run  on  account  of  the  screw-threads  not 
being  of  uniform  value.  In  this  case  the  value  of  each  revolu- 
tion of  the  screw  is  determined  independently,  these  values 
tabulated,  and  the  correction  for  run  from  this  source  deter- 
mined for  any  given  reading.  Again,  as  the  microscope  re- 
volves around  the  limb  with  the  alidade,  the  plane  of  the 
graduations  may  not  remain  at  a constant  distance  from  the 


GEODETIC  SURVEYING. 


483 


objective,  in  which  case  the  size  of  the  image  would  vary  to  a 
corresponding  degree.  To  determine  this,  the  values  of  ten- 
minute  spaces  are  determined  on  various  parts  of  the  limb, 
and  if  these  are  not  constant,  then  a table  of  corrections  for  run 
may  be  made  out  for  different  parts  of  the  circle. 

For  reading  on  graduated  straight  lines  the  double  threads 
give  better  results  than  either  the  single  thread  or  the  inter- 
secting threads.  The  space  between  the  threads  should  be  a 
little  greater  than  the  width  of  the  image  of  the  graduation- 
line, so  that  a narrow  strip  of  the  limb’s  illuminated  upper 
surface  may  appear  on  either  side  of  the  graduation  and  inside 
the  wires.  The  setting  is  then  made  so  as  to  make  these  illu- 
minated lines  of  equal  width.  It  is  conceded  that  such  an  ar- 
rangement will  give  more  exact  readings  than  any  other  that 
has  been  used. 

The  magnifying  power  of  the  microscope  is  from  thirty  to 
fifty. 

358.  Programme  of  Observations. — There  are  two  gen- 
eral methods  of  reading  angles  in  triangulation  work.  One 
method  consists  in  measuring  each  angle  inde- 
pendently, usually  by  repeating  it  a number  of 
times  by  successive  additions  on  the  limb,  and 
then  reading  this  multiplied  angle,  which  is  di- 
vided by  the  number  of  repetitions  to  give  the  o 
true  value  of  the  angle.  In  the  other  method 
the  readings  are  made  on  the  several  stations  in 
order,  as  A,  B,  C,  D,  and  Ey  in  the  figure,  and 
the  angles  found  by  taking  the  difference  between 
the  successive  readings.  Each  method  has  its 
advantages  and  disadvantages.  If  the  instrument  has  an  ac- 
curate fitting  in  the  axis,  clamps  which  can  be  set  and  loosened 
without  disturbing  the  positions  of  the  plates,  is  provided  with 
verniers  which  have  a coarse  reading,  as  twenty  or  thirty  sec- 
onds, and  accurate  work  is  desired,  and  if  such  an  instrument 


484 


SUR  VE  YING. 


is  mounted  on  a low,  firm  station,  then  the  method  by  repeti- 
tion would  give  superior  results.  If  any  of  these  conditions  are 
not  fulfilled,  and  especially  if  the  instrument  is  provided  with 
micrometer  microscopes,  whereby  readings  may  be  taken  to 
the  nearest  second  of  arc,  it  is  much  more  convenient,  cheaper, 
and  generally  more  accurate  to  read  the  stations  continuously 
around  the  horizon,  back  and  forth,  until  a sufficient  number 
of  readings  have  been  obtained. 

359.  The  Repeating  Method. — This  method  was  for- 
merly used  almost  exclusively,  but  the  other  is  the  only  one 
now  used  with  the  most  accurate  instruments.  It  was  found 
that  systematic  errors  were  introduced  in  the  method  by 
repetition  of  a single  angle,  due  largely  to  the  clamping  appa- 
ratus. If  this  method  is  used  the  repetitions  should  be  made 
first  towards  the  right  and  then  towards  the  left ; the  number 
of  repetitions  making  a set  should  be  such  as  to  make  the  mul- 
tiplied angle  a multiple  of  360°,  as  nearly  as  possible,  so  as  to 
eliminate  errors  of  graduation  on  the  limb.  Thus,  for  an  angle 
of  60°  repeat  it  six  times  and  then  read.  For  the  second  set 
repeat  six  times  in  the  opposite  direction,  and  with  telescope 
inverted.  If  triangulation  work  is  to  be  done  with  the  ordi- 
dary  engineer’s  transit,  which  reads  only  to  30  seconds  or  one 
minute,  this  method  may  give  very  fair  provided  there 

is  no  movement  of  circles  from  the  use  of  the  clamping  apparatus 
and  no  lost  motion  in  the  axes.  The  programme  would  be  as 
follows : 


PROGRAMME. 

Telescope  Normal. 

Set  on  left  station,  and  read  both  verniers. 
Unclamp  above  and  set  on  right  station. 


below 

above 

below 

above 

etc., 


left 

right 

left 

right 

etc., 


GEODETIC  SURVEYING. 


485 


until  the  entire  circle  has  been  traversed,  then  read  both  ver- 
niers while  pointing  to  right  station.  The  total  angle  divided 
by  the  number  of  repetitions  is  the  measure  of  the  angle 
sought. 


I. 


2. 


3* 


5- 

6. 


Telescope  Reversed. 

Set  on  right  station,  and  read  both  verniers. 
Unclamp  above  and  set  on  left  station. 

“ below  “ ‘ 


above 

below 

above 

etc.. 


right 

left 

right 

left 

etc.. 


until  the  entire  circle  has  been  traversed  by  each  vernier,  when 
both  verniers  are  read  on  the  left  station. 

The  repetition  in  opposite  directions  is  designed  to  elimi- 
nate errors  from  clamp  and  axis  movements,  and  the  revers- 
ing of  the  telescope  is  designed  to  eliminate  errors  arising 
from  the  line  of  sight  not  being  perpendicular  to  the  horizon- 
tal axis,  and  from  the  horizontal  axis  not  being  perpendicular 
to  the  vertical  axis  of  the  instrument.* 

As  many  such  sets  of  readings  may  be  made  as  desired, 
but  there  should  always  be  an  even  number,  or  as  many  of  one 
kind  as  of  the  other.  It  will  be  observed  that  two  pointings 
are  taken  for  each  measurement  of  the  angle,  but  compara- 
tively few  readings  are  made. 

360.  Method  by  Continuous  Reading  around  the  Hori- 
zon.— By  this  method  the  limb  is  clamped  in  any  position,  and 


* In  case  the  instrument  used  is  a theodolite,  and  its  telescope  cannot  be 
revolved  on  its  horizontal  axis,  it  should  be  lifted  from  the  pivot  bearings  and 
turned  over  end  for  end,  leaving  the  pivots  in  their  former  bearings.  If  this 

cannot  be  done  conveniently,  then  the  limb  should  be  shifted  by  (see  next 

ft 

page)  each  time,  and  this  will  result  in  mostly  eliminating  these  same  errors  of 
collimation  and  inclination  of  horizontal  axis 


486 


SUR  VE  YING. 


left  undisturbed  except  between  tlie  different  sets  of  readings. 
The  pointings  are  made  to  the  stations  in  succession  around 
the  horizon,  and  both  verniers,  or  microscopes,  read  for  each 
pointing.  Thus,  if  the  instrument  were  at  o,  h'ig.  137,  the 
pointings  would  be  made  to ‘'in d E.  If  the  telescope 
is  now  carried  around  to  the  right  until  the  line  of  sight  again 
falls  on  A,  and  a reading  taken,  the  observer  is  said  to  close 
the  horizon  ; that  is,  he  has  moved  tlie  telescope  continuously 
around  in  one  direction  to  the  point  of  beginning.  If  the  two 
readings  here  do  not  agree,  the  error  is  distributed  among  the 
angles  in  proportion  to  their  number,  irrespective  of  their  size. 
It  is  questionable  whether  such  an  adjustment  adds  much  to 
the  accuracy  of  the  angle  values,  and  therefore  it  is  common 
to  read  to  the  several  stations  back  and  forth  without  closing 
the  horizon.  Sum-angles  can  afterwards  be  read  if  desired. 
Thus,  after  the  regular  readings  have  been  taken  on  the  sta- 
tions, the  angle  AOE,  or  AOC,  and  COE,  may  be  read,  and  so 
one  or  more  equations  of  condition  obtained. 

If  the  station  is  tall,  there  is  always  a twisting  of  its  top  in 
clear  weather  in  the  direction  of  the  sun’s  movement.  This 
twisting  effect  has  been  observed  to  be  as  much  as  1"  in  a, 
minute  of  time  on  a seventy-five-foot  station.  To  eliminate 
this  action  the  readings  are  taken  both  to  the  right  and  to  the 
left.  The  reading  of  opposite  verniers,  or  microscopes,  elimi- 
nates errors  of  eccentricity,  the  inverting  of  the  telescope  elimi- 
nates errors  of  adjustment  in  the  line  of  collimation  and  hori- 
zontal axis,  and  to  eliminate  periodic  errors  of  graduation  each 
angle  is  read  on  symmetrically  distributed  portions  of  the  limb. 
To  accomplish  this  the  limb  is  shifted  after  each  set  of  read- 

I So^ 

ings  an  amount  equal  to  ,*  where  n is  the  number  of  sets 
of  readings  to  be  taken.  The  following  is  the 


For  exception,  see  foot-note  on  previous  page. 


GEODETIC  SURVEYING. 


487 


PROGRAMME. 


1ST  Set. 


Telescope  normal. 
Read  to  right. 
Read  to  left. 
Telescope  inverted. 
Read  to  right. 


Read  to  left. 
Shift  the  Limb. 


2D  Set. 


Telescope  inverted. 
Read  to  right. 
Read  to  left. 
Telescope  normal. 
Read  to  right. 
Read  to  left. 
Shift  the  Limb. 


Evidently  each  set  is  complete  in  itself,  and  as  many  com- 
plete sets  may  be  taken  as  desired,  but  no  partial  sets  should 
be  used.  If  the  work  is  interrupted  in  the  midst  of  one  set  of 
readings,  the  partial  set  of  readings  should  be  rejected,  and 
when  the  work  is  resumed  another  set  begun.  In  reducing  the 
work,  if  one  reading  of  any  angle  is  so  erroneous  as  to  have  to 
be  rejected  this  should  vitiate  that  entire  set  of  readings  of 
that  angle. 

If  preferred,  the  telescope  may  be  inverted  between  the 
right  and  left  readings,  and  then  two  readings  on  each  mark 
would  constitute  a complete  set,  when  the  limb  could  be 
shifted  again.  If  this  were  done,  the  readings  at  o.  Fig.  137, 
would  be : 

1ST  Set  \ Telescope  Normal — Read  ABODE. 

( “ Inverted  “ EDCBA. 

% 

Shift  the  Limb. 

2D  Set  i Telescope  Inverted — Read  ABODE, 

\ “ Normal  “ EDOBA. 

Shift  the  Limb. 

361.  Atmospheric  Conditions. — In  clear  weather  not  even 
fair  results  can  be  obtained  during  the  greater  part  of  the  day. 
From  sunrise  till  about  four  o’clock  in  the  afternoon  in  sum- 
mer the  air  is  so  unsteady  from  the  heated  air-currents  th^»- 


488 


SURVEYING. 


any  distant  target  is  either  invisible  or  else  its  image  is  so  un- 
steady as  to  make  a pointing  to  it  very  uncertain.  From 
about  four  o’clock  till  dark  in  clear  weather,  and  all  day  in 
densely  cloudy  weather  with  clear  air,  good  work  can  be  done. 
If  heliotropes  are  used,  the  work  is  limited  to  clear  weather. 
It  has  often  been  proposed  to  do  such  work  at  night,  but  the 
lack  of  a simple  and  efficient  light  of  sufficient  strength  has 
usually  prevented.  The  higher  the  line  of  sight  above  the 
ground  the  less  it  is  affected  by  atmospheric  disturbances. 

362.  Geodetic  Night  Signals. — Mr.  C.  O.  Boutelle,  of  the 
U.  S.  Coast  and  Geodetic  Survey,  made  a series  of  experiments 
in  1879  Sugar  Loaf  Mountain,  Maryland,  for  the  purpose  of 
testing  the  efficiency  of  certain  night  signals  and  the  compara- 
tive values  of  day  and  night  work.  His  report  is  given  in  Ap- 
pendix No.  8 of  the  Report  of  the  U.  S.  C.  and  G.  Survey  for 
1880.  It  seems  that  either  the  common  Argand  or  the  “ Elec- 
tric” coal-oil  lamp,  assisted  by  a-  parabolic  reflector  or  by  a 
large  lens,  gives  a light  visible  for  over  forty  miles.  His  con- 
clusions are : 

1.  That  night  observations  are  a little  more  accurate  than 
those  by  day,  but  the  difference  is  slight. 

2.  That  the  cost  of  the  apparatus  is  less  than  that  of  good 
heliotropes. 

3.  That  the  apparatus  can  be  manipulated  by  the  same  class 
of  men  as  those  ordinarily  employed  as  heliotropers.  , 

4.  That  the  average  time  of  observing  in  clear  weather  may 
be  more  than  doubled  by  observing  at  night,  and  thus  the  time 
of  occupation  of  a station  proportionately  shortened ; “clear- 
cloudy”  weather,  when  heliotropes  cannot  show,  can  be  utilized 
at  night. 

363.  Reduction  to  the  Centre. — It  sometimes  happens 
that  the  instrument  cannot  be  set  directly  over  the  geodetic 
point,  as  when  a tower  or  steeple  is  used  for  such  point.  In 
this  case  two  angles  of  each  of  the  triangles  meeting  here  may 


GEODETIC  SURVEYING. 


489 


be  measured  and  the  third  taken  to  be  180°  minus  their  sum, 
or  the  instrument  maybe  mounted  near  to  the  geodetic  point 
and  all  the  angles  at  this  station  measured  from  this  position. 
These  angles  will  then  be  very  nearly  the  same  as  though 
measured  from  the  true  position,  and  may  readily  be  reduced 
to  what  they  would  have  been  if  the  true  station  point  had 
been  occupied.  Thus  in  Fig.  138  let  C be  the  true  station  to 
which  pointings  were  taken  from  other  stations,  and  C the  posi- 
tion of  the  instrument  for  measuring  the  angles  at  this  station. 
The  line  AB  is  a side  of  the  system  whose 
length  has  been  found.  From  the  measured 
angles  at  A and  B the  approximate  value  of 
the  angle  C is  found  and  the  lengths  of  the 
sides  a and  b computed.  At  C the  angle 
AC’ B is  measured  with  the  same  exactness 
as  though  it  were  the  angle  C itself  and  the 
angle  CC ' B — a is  measured  by  a single  ob- 
servation. The  distance  CC'  = r is  also  a 
found.  Since  the  exterior  angle  at  the  inter-  Fig.  138. 

section  d.sADB,  is  equal  to  the  sum  of  the  opposite  interior 
angles,  we  have 

C-{-j^=C'-\-x,  or  C=C'  + {x~j/).  . . (i) 

In  the  triangle  ACC  we  have  the  sides  b and  r and  the 
angle  ^ 6"^  ^7  known,  whence 

rsin{C'-\-a) 

sm  ;i:  = ^ ; 

b 

k . , . . . (2) 

. r sm  a 

similarly  sm/  = — ^ — . 

Since  x and  / are  very  small  angles,  their  sines  are  propor- 
tional to  their  arcs,  and  we  may  write  sin  x — x sin  where 


490 


SUR  VE  Y I NG. 


X is  expressed  in  seconds  ; similarly  sin  ^ sin  i",  and  equa- 
tions (2)  become 

r sin  {C'  -j- 
b sin  i"  ’ 


X — 


y = 


r sin  a 
a sin  i"‘ 


Substituting  these  values  in  (i)  we  have 

r /sin  C-\-  a sin 
^ ^ ^ sini"  \ ~ b a 


(3) 


• (4) 


where  the  correction  to  C is 


But  since  the  angles  x and 
equal  to  their  arcs,  and  we  ha 


given  in  seconds  of  arc.  The 
signs  of  the  trigonometrical 
functions  of  the  angle  a must 
be  carefully  attended  to,  as  it  is 
measured  continuously  from  B 
around  to  the  left  to  360°. 

The  following  is  another  so- 
lution of  the  same  problem : 
Measure  the  perpendiculars 
from  C upon  AC  and  BC,  Fig. 
139,  calling  them  m and  n re- 
spectively. Then  from  equation 
(i)  above  we  have 

C=C+{x-  y). 

y are  very  small,  their  sines  are 
^e,  in  seconds  of  arc, 


m 


X = 


whence 


b sin  i" 

C=C 


and 


y = -7; 


a sin  I 


I (m  ii\ 

ihrF'  \Jb  ~ hr 


(5) 


GEODETIC  SURVEYING. 


491 


There  are  four  cases  corresponding  to  the  four  positions  of 
C\  as  shown  in  Fig.  140.  For  these  several  cases  we  have 


C=  c: 

I 

hn 

n\ 

sin  i" 

\b 

a)  ’ 

1 

11 

I 

(m 

n\ 

sin  i" 

\J 

-a!' 

C = C3'  + 

I 

(m 

sin  i" 

11 

1 

I 

(m 

sin  i" 

\d 

J 

ADJUSTMENT  OF 

THE  MEASURED  ANGLES. 

364.  Equations  of  Conditions. — When  any  continuous 
quantity,  as  an  angle  or  a line,  is  measured,  the  observed  value 
is  always  affected  by  certain  small  errors.  Indeed,  it  would 
not  be  possible  even  to  express  exactly  the  value  of  a contin- 
uous quantity  in  terms  of  any  unit,  as  degrees  or  feet  and 
fractional  parts  of  the  same,  even  though  this  value  could  be 
exactly  determined.  If,  therefore,  the  measured  values  of  the 
three  angles  of  a triangle  be  added  together,  the  sum  will  not 
be  exactly  180°.  But  we  know  that  a rigid  condition  of  all  tri- 
angles is  that  the  sum  of  the  three  angles  is  180°.  An  equation 
which  expresses  a relation  between  any  number  of  observed 
quantities  which  of  geometrical  necessity  must  exist  is  called 
an  equation  of  condition,  or  a condition  equation.  Thus,  in 
the  above  case,  if  A' , B\  and  C be  the  mean  observed  values 
of  the  angles,  and  A,  B,  and  C their  true  values,  we  would 
have  for  our  condition  equation 


A + B+C=  180* 


(>) 


492 


S UR  VE  YING. 


We  would  also  have 

A'+a'  = = C'-^c=C, 

where  a\  b' , and  c'  are  small  corrections  to  the  measured 
B values  A',  B\  and  C'  which  are  to  be 

determined. 

Let  us  suppose  that  the  length  of 
the  side  b has  been  exactly  meas- 
ured,* then  when  the  true  values  of 
the  angles  are  found  we  may  com- 
pute the  other  two  sides.  If  the  sides 
b and  c have  both  been  measured,  the 
length  of  the  side  c as  computed  from  b must  agree  with  its 
measured  length,  and  so  we  might  write  the  condition  equation 


b sin  {C'+  c') 
~~  sin  {B'  + b')  ' 


(2) 


Again,  if  the  side  a had  been  measured  and  its  exact  length 
found,  we  would  obtain  the  third  condition  equation, 

_ b sin  {A'  + a') 

^ ~~  sin  {B'  -f-  b') 

We  now  have  three  independent  equations  involving  three 
unknown  quantities,  and  can,  therefore,  find  the  quantities  a\ 
b\  and  c' . But  if  only  one  side  had  been  measured,  we  should 
have  had  but  one  equation  from  which  to  determine  three  un- 
known quantities.  Evidently  there  is  an  infinite  number  of 


* This  assumption  is  made  in  regard  to  the  measured  base-lines  in  a trian- 
gulation-system, since  its  exactness  is  so  much  greater  than  can  be  obtained 
in  the  angle-measurements. 


GEODETIC  SURVEYING. 


493 


sets  of  values  of  a’ , b\  and  c\  which  would  satisfy  this  equation. 
If  we  now  impose  the  condition  that  the  corrections  shall  be 
the  most  probable  ones,  then  there  is  but  one  set  of  values  that 
can  be  taken. 

Equation  (i)  is  called  an  angle  equation^  since  only  angles 
are  involved ; while  equations  (2)  and  (3)  are  called  side  equa- 
tions^ since  the  lengths  of  the  sides  are  also  involved. 

365.  Adjustment  of  a Triangle. — The  finding  and  ap- 
plying of  the  most  probable  corrections  to  the  measured  values 
of  the  angles  of  a system  of  triangulation  is  called  adjusting 
the  system.  In  the  case  of  a single  triangle  with  one  known 
side  and  three  measured  angles,  we  have  seen  that  there  is  but 
one  equation  of  condition.  If  the  three  angles  have  been; 
equally  well  observed,  then  it  is  most  probable*  that  they  are 
all  equally  in  error,  and  hence  this  condition  of  highest  proba.-. 
bility  gives  us  the  probability  equation 

a'  = b'  = c' (4) 

which  enables  the  corrections  to  be  determined. 

Thus,  let  A'  + C - iSo^  = a'  + b'  + c' = ' 

then  from  (4)  we  have 


where  V is  the  error  of  closure  of  the  triangle. 


• • (5) 


* That  is,  this  relation  is  more  probable  than  are  any  other  smg/^  relation 
that  can  be  assigned,  but  of  course  it  is  not  more  probable  than  all  other  eases 
combined. 


494 


SUR  VE  YING. 


ADJUSTMENT  OF  A QUADRILATERAL. 

366.  The  Geometrical  Conditions. — In  the  quadrilateral 
in  Fig.  142  there  are  eight  observed  angles,^,,  /i,,  etc. 
The  geometrical  conditions  which  must  here  be  fulfilled  are  : 

{a)  The  sum  of  all  the  angles  of  any  triangle  must  be  180° 
plus  the  spherical  excess*  and  the  opposite  angles  at  the 
intersection  of  the  diagonals  must  be  equal. 

(b)  The  computed  length  of  any  side,  as  DC^  when  obtained 
from  any  other  side,  as  AB,  through  two  independent  sets  of 
triangles,  as  ABC,  BDC,  and  ABD,  ADC,  shall  be  the  same  in 
both  cases. 

The  probability  condition  is  that  the  set  of  corrections  ap- 
plied to  the  several  angles  shall  be  more  probable  than  any 
other  one  of  the  infinite  number  of  sets  of  corrections  which 
would  satisfy  the  other  condition. 

The  condition  given  in  {a)  gives  rise  to  the  angle  equations, 
and  that  given  in  {p)  gives  one  side  equation. 

There  are  evidently  eight  unknown  corrections  to  be  de- 
termined. 

367.  The  Angle-equation  Adjustment. — In  the  quadri- 
lateral ABCD  we  have  four  triangles  in  which  all  the  angles 
have  been  observed,  two  sets  of  opposite  angles  where  the 
other  two  angles  of  the  corresponding  triangles  have  been  ob- 
served, and  the  quadrilateral  itself  in  which  all  the  angles  have 

*It  is  not  necessary  to  take  account  of  the  spherical  excess  in  computing  a 
single  triangle  or  quadrilateral  ; but  if  azimuth  is  to  be  carried  over  a series  of 
triangles  it  is  necessary  that  all  the  angles  be  spherical  angles.  In  this  place 
spherical  excess  will  be  omitted  ; but  if  it  is  desirable  to  introduce  it,  it  is  in- 
serted in  equations  (i),  (2),  and  (3),  the  right  members  then  becoming  A <?i.  A 
-f-  e-x,  and  A -f-  e^,  where  is  the  residual  excess  of  the  angle  A OB  over  that  of 
the  angle  DOC  (being  negative  in  this  case),  e-i  is  the  excess  of  angle  B OC  o\tr 
that  of  the  angle  AOD,  and  <?3  is  the  spherical  excess  for  the  entire  quadri- 
lateral. The  spherical  excess  may  be  taken  as  i"  for  each  75  square  miles  of 
area,  and  this  is  to  be  divided  equally  amongst  the  angles  of  the  figure.  The 

206000 

formula  for  spherical  excess  is  E (in  seconds)  = j — , where  A is  area  in 

square  miles,  and  r is  radius  of  the  earth  in  miles. 


GEODETIC  SURVEYING. 


495 


been  observed  ; making-,  in  all,  seven  geometric  conditions  to 
be  fulfilled.  Only  three  of  these  conditions  are  independent, 
however,  since  where  any  three  independent  conditions  are 
fulfilled  the  remaining  four  are  fulfilled  also.  Thus,  a great 
variety  of  conditioned  equations  could  be  formed,  but  we  will 


D 


C 


K 


B 


Fig.  142. 


choose  the  three  which  give  the  simplest  equations,  viz. : that 
the  opposite  central  angles  shall  be  equal,  and  that  the  sum  of 
all  the  angles  of  the  quadrilateral  shall  be  360°.  These  give 
rise  to  the  following  equations  : 

If  Aj,  ^2,  C^,  etc.,  be  the  observed  angles,  and  /j,  4,  and  ^ 

the  residuals  in  the  several  combinations,  due  to  erroneous 
determinations,  then  we  have : 


i8o°— (^,+^2)  — 1 180  — (6’b  + — /„ 


or 


- ^ -^2 


+ ^6  + ^6 


Similarly  —B^— 


+ A + ^ 


= k.  (2) 


and  ^,+^,+4^3+  6^,  + + A + A + - 360°  - /,  (3) 


496 


SUR  VE  YING. 


If  the  angles  have  all  been  equally  well  observed, — that  is,  if 
their  mean  observed  values  have  equal  credence, — then  they 
are  said  to  have  equal  weight,  and  any  residual  arising  from 
any  combination  of  angles  should  be  distributed  uniformly 
among  the  angles  forming  such  combination/'^  1 hus  /,  arises 
from  the  angles  and  This  residual  should  there- 

fore, be  divided  equally  between  these  four  angles.  When 
this  is  done  we  have 

+ + + = . (4) 

Similarly 

-(*+5)-(<r.+^)  + A-4  + ^.-^  = o.  . (5) 

It  is  evident  that  if  4 be  now  divided  uniformly  among  the 
eight  observed  angles,  it  will  not  affect  the  two  adjustments 
already  made ; neither  have  the  adjustments  already  made 
affected  the  third  condition,  expressed  by  eq.  (3),  since  equal 
amounts  have  been  added  and  subtracted.  Hence  these  ad- 
justments may  be  made  in  sequence  as  well  as  simultaneously, 
and  we  shall  have  for  the  total  corrections  for  angle-equa- 
tions 


A,-  1 

<1,  // 
^8  4) 

^ ^2  — 

f4  4 ^ 

\8  4/ 

+ 

1 

1 

l+c 

— 

(1,  V 
Is  4> 

) + ^5  — 

(s'  + ^Z 

i+a-(|+4) 

+ A 

- 1 

^8 

1 + ^8  ~ 

('f+f: 

0 

11 

0 

CO 

1 

...  (6) 

* The  errors  in  the  mean  observed  values  of  the  angles  are  supposed  to  re- 
sult from  the  small  incidental  errors  and  approximations  made  in  pointing, 


GEODETIC  SURVEYING. 


497 


Or  if  Vs,  etc.,  be  the  total  corrections  to  the  several  mean 

observed  angles  for  angle-equations,  we  have 


4-2/, 

v,  = Vs= g , 


4 - 24 
8 ’ 


V,  = V,r= 

V,  — Vs  - 


4 + 2/^ 

8 ’ 

4 24 

8~’ 


(7) 


368.  The  Side-equation  Adjustment. — In  the  quadri- 
lateral shown  in  the  figure,  let  AB  be  the  known  side,  and  CB 
the  required  side,  which  is  to  be  computed  through  two  inde- 
pendent sets  of  triangles.  Let  A/,  BJ,  etc.,  be  the  several 
angles  corrected  for  angle  conditions  by  the  corrections  found 
in  eq.  (7). 

As  computed  through  the  first  set  of  triangles,  we  have 


DC  = 


BC  sin  Bs 
sin  Ds 


AB  sin  A^  sin  B^ 
sin  sin 


(8)  • 


Similarly 


DC  ^ 


AD  sin  As 
sin  Cs 


AB  sin  Bs  sin  A^ 
sin  Cs  sin  D^ 


(9) 


Whence 


sin  sin  B^  _ sin  sin  A^ 

sin  Cl  sin  Dl  ~ sin  Cl  sin  Dl  ’ 


reading,  etc. ; in  other  words,  they  are  supposed  to  be  errors  of  observation  and 
not  instrumental  errors,  these  latter  having  been  eliminated  by  the  method  of 
making  the  observations.  Since  the  sources  of  the  errors  of  observation  are 
the  same  for  small  as  for  large  angles,  it  follows  that  they  should  be  credited 
with  equal  portions  of  the  aggregate  error  of  any  combination  of  angles,  re- 
gardless of  the  size  of  the  angles  themselves. 

32 


498 


SC/A'  VE  YING. 


or 


sin  sin  sin  sin  Z?/  _ 

sin  sin  sin  sin  ~~  ’ 


which  is  called  the  side-equation. 

It  is  evident  that  in  any  case  where  the  angles  have  all 
been  observed,  even  after  they  have  been  adjusted  for  the 
angle-conditions,  this  equation  will  not  hold  true,  the  value  of 
the  left  member  being  a little  more  or  less  than  one.  When 
put  into  the  logarithmic  form  for  computation,  therefore,  we 
will  have 


log  sin  A^  log  sin  + log  sin  + ^^g  sin  D^' 

—'log  sin  B^  — log  sin  — log  sin  Z?/  — log  sin  A^  = /^,  (i  i) 

where  ^ is  the  logarithmic  residual  due  to  erroneous  observa- 
tions. 

We  must  now  distribute  this  residual  among  the  log  sines 
according  to  the  most  probable  manner  of  the  occurrence  of 
the  errors  which  caused  it.  For  a given  small  error,  as  i",  in 
any  angle,  the  effect  on  the  log  sine  is  measured  by  the  loga- 
rithmic tabular  difference  for  for  that  angle.  This  tabular 
difference  varies  for  different  angles,  being  large  for  angles 
near  zero  or  i8o°,  and  small  for  angles  near  90°. 

Let  7^/,  7^/,  , etc.,  be  the  corrections  to  be  made  to  the 

angles  A^,  B^,  B^',  etc.,  for  the  side-equation  (ii),  and  let 

<^3,  etc.,  be  the  corresponding  logarithmic  tabular  differences 
for  i". 

Now,  the  influences  on  4 of  the  small  angular  errors  were 
in  direct  proportion  to  the  tabular  differences  of  the  correspond- 
ing log  sines  ; therefore  the  corrections  should  be  in  proportion 
to  the  corresponding  tabular  differences.  In  other  words,  the 


GEODETIC  SURVEYING. 


499 


corrections  are  weighted  in  proportion  to  their  tabular  differ- 
ences.* We  therefore  have  the  numerical  relation: 


\ d^w  ::  etc., 


or,  paying  attention  to  signs, 


d’^  d^  d^ 


(12) 


But  since  the  log-sine  correction  is  the  angular  correction 
multiplied  by  the  tabular  difference,  and  since  the  sum  of  these 
would  equal  4,  we  would  have 


v^d,—v^d^-\-v^d,-v^d.-{-v,'d,-v'd.-\-v,'d,-v^d,=  — l..  . (13) 

From  equations  (12)  and  (13)  we  are  to  find  the  side-equation 
corrections  7/3',  etc. 

Dividing  eq.  (13)  by  eq.  (12),  term  by  term,  we  have 


+ 


+ + + + 


d: 


I//  + V,' 


-^-  = +etc. 


* To  illustrate  this  principle  more  fully,  let  us  suppose  that  for  a ^change 
of  i"  in  the  angles  Ax  and  the  corresponding  changes  in  the  log  sines  are  i 
for  Ax  and  2 for  A-i\  then  for  a given  error  of  i in  log  sin  Ax  -j-  log  sin  A-i  — I 
there  are  two  chances  that  it  came  from  to  one  chance  that  it  came  from  Ax, 
when  these  angles  were  equally  well  observed.  If  the  error  is  to  be  divided 
between  the  angles  Ax  and  y^a,  therefore,  we  should  make  the  correction  to  y^a 

i • y , , . Vx  V<i 

twice  as  great  as  the  correction  to  Ax,  ov  Vx  v-x  dx  d-x,  whence  — — 

dx  — d-x' 

The  same  reasoning  would  hold  evidently  for  any  number  of  angles,  hence 
equation  (12). 


500 


SURVEYING. 


Whence  we  have,  for  the  values  of  these  corrections, 


d^  d^  d^  * d.j 


('4) 


We  have  now  found  a set  of  corrections,  t/,,  e/,,  etc.  (eq. 
7),  for  the  angle-equations,  and  a set  of  corrections,  v^',  ^ ^ 

etc.  (eq.  14),  for  the  side-equation  ; but  they  were  determined 
independently  and  not  simultaneously,  and  therefore,  when 
successively  applied,  each  set  of  corrections  will  disturb  the 
former  adjustment  somewhat.  Thus,  if  the  corrections  in  eq. 
(7)  be  first  applied,  and  then  those  of  eq.  (14),  using  the  par- 
tially corrected  angles  in  finding  by  eq.  (ii),  we  would  find 
eq.  (10)  would  be  satisfied,  but  /„  4,  and  4,  in  equations  (i),  (2), 
and  (3),  would  now  not  be  zero  when  the  newly  adjusted  angles 
were  used.  Another  set  of  corrections  z//',  v",  v"y  etc.,  might 
now  be  found  by  eq.  (7)  for  the  adjusted  angles  A/', 
etc.,  and  so  on  by  successive  approximations,  using  the  correc- 
tions of  equations  (7)  and  (14)  alternatefy,  until  both  sets  of 
conditions  were  satisfied  within  the  desired  limits.  It  will 
usually  be  found,  however,  that  the  adjustment  for  side-equa- 
tion does  not  materially  disturb  that  for  angle-equations.  If 
the  angles  were  all  the  same  size,  so  that  the  corrections  to  the 
log  sines  would  have  equal  weight,  the  first  adjustment  would 
remain  undisturbed.  In  this  case,  the  corrections  for  side- 
equation  would  all  be  numerically  equal,  the  odd  and  even 
subscripts  having  opposite  signs.  If  the  observed  angles  range 
between  30°  and  60°,  as  they  would  in  a fairly  symmetrical 
quadrilateral,  then  the  errors  of  this  approximation  would  be 
quite  inappreciable. 


GEODETIC  SURVEYING, 


501 


369.  Rigorous  Adjustment  for  Angle-  and  Side-equa- 
tions.— Let  the  angle-equation  adjustments  be  applied  as  given 
by  eq.  (7).  Then,  using  these  adjusted  angles,  let  the  correc- 
tions to  the  angles  for  side-equation  be  so  expressed  that  they 
shall  not  be  inconsistent  with  the  angle-equation  conditions, 
whatever  their  values.  This  may  be  done  by  letting 


t/  = 

■^0  + ^1, 

^0  + ^3  i 

< = 

^0  — -^3 ; 

^/  = 

— -^0  + -^'2^ 

— -^0  + -^4 ; 

T'/  — 

1 

1 

1 

1 

Then,  analogous  to  eq.  (13),  we  may  write 


+ X,)  — dlx,  - X,)  — d,(x,  - X,}  + d,(x,  + x,) 
+<(x,+x,)-d,(x,-x,)-d,(x„—x,)-i-dXx,-{-x,)^-/,;  . (i6) 


or 


(^1  — ^2  “ ^3  + ^4  + ^6  ” + ^8)-^0  + W + 

+ (^3  + ^4)-^a  + (^6  + ^6)^8  + (^7  + <^8)^4  = ~ * (l?) 

wherein  is  given  by  eq.  (ii),  and  the  <3^’s  are  the  tabular 
differences  for  one  second  for  the  several  log  sines  as  before. 
If,  for  simplicity,  we  write  for  the  coefficients  of  ,^0, 
and  jtr^,  respectively,  C^,  C,,  C^,  and  then  (17)  becomes 

+ ^1^1  “h  ^2-^2  + ^8*^3  + C^4  = — ^4.  . . (18) 

It  now  remains  to  find  the  values  of  ;ir,,  and  at^,  such 

that  their  combinations  which  make  up  the  angle-corrections  as 
given  in  eqs.  (15)  shall  be  the  most  probable. 


502 


SURVEYING. 


To  make  (i8)  symmetrical  with  (15),  we  may  put  it  in  the 
following  form : 

^0  + ”^0  + + (“  ^0 

+ {~^o+C,x^  = - /.•  • (19) 

By  a line  of  reasoning  exactly  similar  to  that  by  which  eq. 
(12)  was  obtained,  we  have 


^.+  X, 


a 


-c. 


whence 


Similarly  with  the  other  equations  of  (15)  and  the  corre- 
sponding coefficients  in  (19),  from  which  we  may  obtain 


4£.o 

a 


c: 


c* 


*As  before,  the  condition  of  greatest  probability  makes  the  corrections  pro- 
portional to  their  coefficients  in  the  fundamental  equation  (19);  hence  we  have 


jTo : xi  ::  — \Ci\ 

4 


or,  by  composition  and  division, 

Xq  -|-  X\  Xq 


xi ..  — - 4- 

4 


Xo-{-  Xi  _ Xo  — Xi 

Co  , ~ ~ 'Co  T 
4 4 

and  similarly  with  aro  and  x^,  Xo  and  Xa,  etc. 


Co 


-Cu 


GEODETIC  SURVEYING. 


503 


Therefore  the  condition  of  the  highest  probability  gives 


4£2_f' 

c„~  c\-  c,  - c,  - c:  • 


. . (20) 


Dividing  (18)  by  (20),  term  by  term,  we  have 


a 


CA 


4*^0 


whence 


^ _ 

c,-  c\-  c\- 1,  “ C - 

From  equation  (21)  the  side-equation  corrections  can  be 
computed,  which  will  not  disturb  the  angle-equation  adjust- 
ment, and  which  are  the  most  probable  corrections  to  the 
several  angle-values. 

The  second  or  rigid  method  will  be  found  much  more  satis- 
factory than  the  method  by  approximations.  The  complete 
adjustment  consists  in  applying  to-  the  mean  measured 
values,  the  corrections  from  angle-equations  given  by  equation 
(7),  and  then  applying  to  these  corrected  angles  the  corrections 
found  by  equation  (21). 

Note. — The  results  obtained  in  the  above  adjustments  are 
identical  with  those  found  by  the  method  of  least  squares,  and 
the  fundamental  principle  by  which  they  are  obtained  is  also 
the  same  as  that  of  least  squares,  viz. ; that  the  arithmetic 


. . (21) 


504 


SURVEYING. 


mean  of  properly  weighted  observations  is  the  most  probable 
result,  and  is  identical  with  that  obtained  by  making  the  sum 
of  the  squares  of  the  corrections  a minimum.  For  least-square 
solutions  of  this  problem,  see  Clarke’s  “ Geodesy,”  pp.  263-6, 
and  Wright’s  “ Adjustment  of  Observations,”  pp.  303-8. 

Example. 

The  following  is  the  numerical  computation  of  the  quadrilateral  shown  in 
the  figure.  AB  is  the  known  side,  and  CD  is  to  be  found.  The  mean  observed 
values  of  the  angles  are  given  in  the  second  column.  The  corrections  for 
angle-equations  are  given  in  the  third  column,  and  are  the  same  for  all  three 
methods  of  solution  given  above.  The  spherical  excess  is  here  applied  only  to 
the  quadrilateral  as  a whole,  or  to  h,  thus  distributing  it  equally  among  the 
several  angles.  This  is  a common  way  of  doing  it,  although  if  the  excess  is 
considerable,  and  the  several  triangles  very  unequal  in  size,  as  is  the  case  here, 
it  should  be  applied  to  the  several  triangles  according  to  their  size,  as  stated  in 
the  foot-note,  p.  495. 

In  columns  7 and  8,  the  corrections  for  side-equation  are  worked  out  by  the 
two  methods  given  to  show  the  relative  results.  Thus,  from  eq.  (14)  we  find 
the  values  of  Vx  , va,  etc.,  for  the  first  approximation.  Applying  these  to  the 
first  corrected  values  in  column  4,  and  again  taking  out  the  values  of  A,  /<», 
and  /s,  for  angle-equation  conditions,  we  find  they  are  not  zero,  but  very  small. 
It  would  probably  be  sufficient  to  work  out  a new  set  of  angle-corrections  by 
eq.  (7),  and  then  consider  the  quadrilateral  adjusted.  In  this  example,  the 
final  values  thus  found  would  then  differ  from  the  final  values  by  the  rigid 
adjustment  by  not  more  than  o".2  for  any  angle. 

If  we  compute  CD  from  AB,  assuming  the  latter  to  be  25,000  feet  in  length, 
we  obtain  88,670.9  ft.  in  passing  through  the  triangles  ABC  and  BCD,  while 
if  we  pass  through  the  triangles  ABD  and  ADC  we  obtain  88671.1  ft.,  a dis- 
crepancy of  0.2  ft.,  and  giving  a mean  value  of  88671.0  ft.  The  discrepancy  of 
0.2  ft.  in  the  two  results  by  the  rigid  solution  results  from  not  computing  the 
corrections  beyond  tenths  of  a second. 

If  simply  a check  on  the  final  corrected  values  is  desired,  it  may  be  obtained 
by  adding  them,  when  their  sum  should  equal  360°  spherical  excess,  or  by 
taking  out  the  log  sines  and  seeing  if  4'  in  eq.  (ii)  is  zero.  In  this  case  it  is 
not  zero,  but  9,  resulting  from  not  carrying  out  the  corrections  beyond  tenths 
of  seconds,  as  mentioned  above. 


COMPUTATION  OF  ANGLE-CORRECTIONS  FOR  A QUADRILATERAL  ADJUSTMENT. 


GEODETIC  SURVEYING. 


505 


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c 

E PiH 


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t/)  V ov 

CQ  -H 


CL--  ~ 
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m 0 fO 


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M o VO  VO 
CO  « 10 


ro  fO 


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+ + + + 


0000 

till 


0000 

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1 ! I I 


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ro  o >0 

+ + + + 


+ + + + 


o^  o>  o>  0 


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VO  \o  00  o 
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ov  O'  ov  O 


3 

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«=  S 

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H S'  ^ 


tx  ro  CO  M 


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I I I 


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1 I I I 


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Iw  S 


00000 

+ + + + + 

!l  II  II  II  II 

H H H 


II  II  II 


^ ^ ;j 


II  II  II  II  II 

c ^ c c c 


ro 


ro 


W Ov  VO  00 
10  M 0 CO 

w in  00  ov 
M m ^ yf 

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M o VO 
CO  N 10 

tv  fO  tv 
M fO 


+ + + 

II  II  II  II 


C CQ  U Q 


CQ  U Q < 


5o6 


SURVEYING. 


ADJUSTMENT  OF  LARGER  SYSTEMS. 

370.  Used  only  in  Primary  Triangulation. — The  simul- 
taneous adjustment  of  all  the  angles  in  an  extended  system  of 
triangles  with  one  measured  base  which  is  taken  as  exact,  is  a 
very  complicated  problem.  The  methods  of  least  squares  must 
here  be  applied,  so  that  a discussion  of  this  problem  belongs 
rather  to  a treatise  on  geodesy  than  to  one  on  surveying.  The 
adjustments  of  a triangle  and  of  a quadrilateral  will  be  found 
sufficient  for  all  secondary  work,  or  such  as  is  intended  to  serve 
only  for  topographical  or  geographical  purposes.  Especially 
is  this  true  if  the  stations  be  so  selected  that  the  observed  lines 
will  form  a series  of  quadrilaterals.  The  adjustment  of  these 
quadrilaterals  by  the  rigid  method  given  above  gives  nearly 
as  good  results  as  could  be  obtained  by  reducing  the  work  as  a 
single  system.  For  a discussion  of  the  least-square  methods 
of  adjustment  of  an  extended  system  of  triangles  the  student 
is  referred  to  “Primary  Triangulation  of  the  U.  S.  Lake  Sur- 
vey,” being  Professional  Papers,  Corps  of  Engineers  U.  S.  A., 
No.  24;  Report  of  the  U.  S.  Coast  and  Geodetic  Survey  for 
1875;  Clarke’s  Geodesy;  and  especially  to  Wright’s  “Adjust- 
ment of  Observations.” 

The  facility  and  accuracy  with  which  base-lines  may  now 
be  measured  by  means  of  long  steel  tapes  will  result  in  actually 
measuring  many  more  lines  than  has  heretofore  been  done,  and 
so  errors  from  angular  measurements  will  not  be  allowed  to 
accumulate  to  any  great  extent.  It  is  not  improbable  that 
geodetic  methods  will  be  materially  influenced  by  this  new 
method  of  accurate  measurement. 

371.  Computing  the  Sides  of  the  Triangles.— After  the 
angles  of  the  system  are  adjusted,  the  sides  of  the  triangles  are 
computed  by  the  ordinary  sine  ratio  for  plane  triangles.  If 
the  system  consist  of  simple  triangles,  then  one  side  is  known 
and  the  other  two  sides  computed  from  it.  In  this  case  there 
is  no  check  on  the  computation  except  what  the  computer 


GEODETIC  SURVEYING. 


507 


carries  along  with  him,  or  what  may  be  obtained  from  a dupli- 
cate computation. 

If  the  system  be  made  up  of  a series  of  quadrilaterals,  then 
the  line  which  is  common  to  two  successive  quadrilaterals  is 
computed  through  two  sets  of  triangles  from  the  previous  known 
side.  Thus  if  the  quadrilateral  of  Fig.  142  be  one  of  a series, 
the  lines  in  common  being  AB  and  CD,  then  AB  is  computed 
in  duplicate  from  the  previous  quadrilateral,  and  the  mean  of 
the  two  results  taken.  In  the  triangle  ABD  compute  AD,  and 
then  in  the  triangle  compute  in  the  triangle 

compute  BC,  and  then  in  the  triangle  BCD  compute  again. 
There  are  thus  obtained  two  independent  values  of  DC,  as 
computed  from  AB.  If  the  adjustment  had  been  exact  these 
values  would  have  agreed  exactly,  but  the  adjusted  angles 
were  computed  only  to  the  nearest  second,  or  tenth  of  a second  ; 
hence  the  two  values  of  DC  will  agree  only  to  a corresponding 
exactness.  If  the  system  be  composed  of  quadrilaterals  and 
the  adjustment  be  made  to  the  nearest  second,  then  the  two 
values  of  DC  would  probably  differ  in  the  fifth  or  sixth  signifi- 
cant figure.  If  the  adjustment  be  made  to  the  nearest  tenth 
of  a second,  and  a seven-place  logarithmic  table  be  used,  then 
the  two  values  of  DC  should  begin  to  differ  in  the  sixth  or 
seventh  place.  Of  course  the  adjusted  values  are  not  the  true 
values  of  the  angles,  but  simply  the  most  probable  values.  If 
the  angles  were  not  accurately  measured  the  adjusted  values 
may  still  be  considerably  in  error,  but  any  such  errors  would 
not  prevent  the  two  values  of  CD  from  agreeing,  since  this 
agreement  is  one  of  the  conditions  which  the  adjustment  is 
made  to  satisfy.  The  disagreement  between  the  two  computed 
values  of  CD  comes  only  from  the  inexactness  of  the  computed 
corrections  to  the  angles,  an  angle,  like  a length,  being  an  in- 
commensurable quantity,  and  hence  some  degree  of  approxi- 
mation is  necessary  in  its  expression.  If  the  true  computed 
values  of  CD  differ  by  more  than  the  amounts  above  signified, 


508 


SUR  VE  YING. 


then  it  is  probable  that  an  error  has  been  made  in  the  com- 
putation. 


LATITUDE  AND  AZIMUTH. 

372.  Conditions. — In  the  methods  here  given  for  obtaining 
latitude,  azimuth,  and  time,  the  instrument  used  may  either  be 
an  ordinary  field  transit  mounted  on  its  tripod,  or  a more  elabo- 
rate altazimuth  instrument,  such  as  shown  in  Figs.  132  and  134. 
The  accuracy  sought  is  only  such  as  is  sufficient  for  topographi- 
cal or  geographical  purposes.  Both  the  field  methods  and  the 
office  reductions  are  of  the  simplest  character;  but  all  large 
errors  are  eliminated,  so  that  the  results  will  be  found  as  accu- 
rate as  it  is  possible  to  obtain  with  anything  less  than  the  regu- 
lar field  astronomical  instruments.  This  higher  grade  of  work 
falls  within  the  sphere  of  the  astronomer  rather  than  of  the 
surveyor. 

373.  Latitude  and  Azimuth  by  Observations  on  Cir- 
cumpolar Stars  at  Culmination  and  Elongation. — When 
latitude  and  azimuth  are  to  be  found  to  a small  fraction  of  a 
minute,  or  as  accurately  as  can  be  read  on  the  instrument  used, 
if  this  be  an  ordinary  field  transit,  the  most  convenient  method 
is  by  means  of  observations  on  circumpolar  stars.  The  observa- 
tion for  latitude  is  made  on  such  a star  when  it  is  at  its  upper 
or  lower  culmination,  since  it  is  then  not  changing  its  altitude, 
and  the  observation  for  azimuth  is  made  at  elongation,  since 
then  the  star  is  not  changing  its  azimuth.  At  these  times  a 
number  of  readings  may  be  taken  on  the  star,  thus  eliminating 
instrumental  constants  by  reversals,  since  a half  hour  may  be 
utilized  for  this  work  without  the  star  sensibly  changing  its 
position  so  far  as  the  use  it  is  serving  is  concerned.  Two  close 
circumpolar  stars  have  been  chosen  whose  right  ascensions 
differ  by  about  five  hours  and  thirty  minutes.  They  therefore 
always  give  a culmination  and  an  elongation  about  thirty  min- 
utes apart.  This  is  very  convenient,  since  it  allows  observations 


GEODE  TIC  S UR  VE  YING. 


509 


to  be  made  for  latitude  and  azimuth  at  one  setting  with  a suf- 
ficient intervening  interval  to  complete  one  set  of  observations 
before  commencing  the  next. 

The  two  stars  selected  are  Polaris  (a  Ursse  Minoris),  which  is 
of  the  second  magnitude,  and  51  Cephei,  which  is  of  the  fifth 
magnitude.  Their  relative  positions  are  shown  in  Fig.  143. 


The  position  of  51  Cephei  may  be  described  with  reference 
to  the  line  joining  “ the  pointers,”  in  the  constellation  of  the 
Great  Bear,  with  Polaris.  Thus,  51  Cephei  is  to  the  right  of 
this  line,  when  looking  towards  the  pole-star  along  the  line,  at 
a distance  of  about  three  times  the  sun’s  disk  from  the  line,  and 
of  about  five  times  the  sun’s  disk  from  Polaris,  in  the  direction 
of  the  pointers. 

In  case  51  Cephei  is  not  visible  to  the  naked  eye,  as  it  may 
not  be  on  moonlight  nights,  or  with  a slightly  hazy  atmos- 
phere, it  may  be  found,  when  near  elongation,  by  the  tele- 
scope, as  follows  : 

Having  carefully  levelled  the  instrument,  turn  upon  Polaris. 
When  5 1 Cephei  is  near  its  eastern  elongation  Polaris  is  near 
its  upper  culminatiorn,  and  when  near  its  western  elongation 
Polaris  is  near  its  lower  culmination.  To  find  51  Cephei  at 
eastern  elongation,  therefore,  after  taking  a pointing  on  Pola- 


510 


SUR  VE  YING. 


ris,  lower  the  telescope  (diminish  the  vertical  angle)  by  about 
one  degree  (if  the  time  is  about  twenty  minutes  before  elonga- 
tion), and  then  turn  off  towards  the  east  about  two  and  a half 
degrees.  This  will  bring  the  cross  wires  approximately  upon 
the  star. 

To  find  it  at  western  elongation,  simply  reverse  these  angles  ; 
that  is,  increase  the  vertical  angle  one  degree,  and  turn  off  to 
the  west  two  and  one  half  degrees. 

The  following  table  gives  the  times  of  the  elongations  and 
culminations  of  these  two  stars  for  1885  latitude  40°,  which 
may  be  used  for  observing  azimuth  and  latitude.  The  times 
given  are  for  the  nights  following  the  dates  named  in  the  first 
column. 


TIMES  OF  ELONGATION  AND  CULMLNATION,  1885. 
LATITUDE,  40°. 


! 

Polaris  (a 

Urs.  Min.). 

51  Cephei. 

Date. 

Elon- 

ga- 

tion. 

Time. 

Cul- 

mina- 

tion. 

Time. 

Elon- 

ga- 

tion. 

Time. 

Cul- 

mina- 

tion. 

Time. 

Jan.  I 

W 

I2’>24"*.6  A.M. 

U 

6**29'®.9  P.M. 

W 

5''48“.3  A.M. 

U 

ii>>58'”.6  P.M. 

Feb.  I 

44 

10  22  .2  P.M. 

L 

4 25  .6  A.M. 

“ 

3 46  .4  “ 

“ 

9 56  .7  “ 

Mar.  I 

44 

8 31  .8  “ 

44 

2 35  -I  “ 

I 56  .1  “ 

“ 

8 6 .4  “ 

April  I 

44 

*6  29  .7  “ 

44 

12  33  .1  “ 

“ 

11  54  .0  P.M. 

“ 

*6  4 -3  “ 

May  1 

E 

*4  36  .6  A.M. 

44 

10  35  .2  P.M. 

“ 

9 55  .9  “ 

L 

4 4 .2  A.M. 

June  I 

ii 

2 37  .0  “ 

44 

8 33  -7  “ 

(( 

*7  53  -9  “ 

“ 

2 2 .2  “ 

July  I 

“ 

12  39  .0  “ 

44 

*6  36  .2  “ 

E 

*6  12  .6  A.M. 

44 

12  4 .2  “ 

Aug.  I 

“ 

10  38  .1  P.M. 

U 

4 32  .8  A.M. 

“ 

4 10  .8  “ 

10  2 .4  P.M. 

Sept.  I 

“ 

8 36  .6  “ 

44 

2 31  -3  “ 

“ 

2 9 .1  “ 

8 0 .8  “ 

Oct.  I 

<( 

*6  38  .9  “ 

44 

12  33  .6  “ 

(( 

12  n .4  “ 

44 

*6  3 .1  “ 

Nov.  I 

W 

4 26  .4  A.M. 

44 

10  31  .7  P.M. 

“ 

10  9 .8  P.M. 

U 

3 59  .5  A.M. 

Dec.  I 

“ 

2 28  .2  “ 

44 

8 33  -5  “ 

8 12  .0  “ 

2 I .8  “ 

* Probably  not  visible  to  the  naked  eye. 


From  the  above  table  it  is  evident  that  both  an  elongation 
and  a culmination  of  one  of  these  stars  can  always  be  obtained. 
For  other  days  than  those  given  in  the  table,  either  inter- 


GEODETIC  SURVEYING. 


51I 


polate,  or  find  by  allowing  3”".94  for  one  day,  remembering 
that  each  succeeding  day  the  elongation  occurs  earlier  by  this 
amount. 

For  other  years  than  1885,  take  from  the  table  the  time  cor- 
responding to  the  given  month  and  day,  and  add  0^.35  for 
each  year  after  1885  ; also. 

Add  if  the  year  is  the  second  after  leap-year. 

Add  2^  if  the  year  is  the  third  after  leap-year. 

Add  3™  if  the  year  is  leap-year  before  March  i. 

Subtract  if  the  year  is  leap-year  after  March  i. 

For  the  first  year  after  leap-year  there  is  no  correction  ex- 
cept the  periodic  one  of  0'".35  per  annum. 

For  other  latitudes  than  40°,  add  0^.14  for  each  degree 
south  of  40°  north  latitude,  or  subtract  o"^.  18  for  each  degree 
north  of  40°  north  latitude  for  Polaris,  and  0^.29  and  o°^39  for 
the  corresponding  correction  for  51  Cephei. 

The  following  table  gives  the  pole  distances  of  Polaris  and 
51  Cephei  for  Jan.  i of  each  third  year  from  1885  to  1930: 


POLE  DISTANCE  (90°  — Declination). 


Star. 

1885. 

1888. 

1891. 

1894. 

1897. 

1900. 

1903. 

1906. 

Polaris. . . . 

51  Cephei.. 

i°i8'i6" 

2 46  35 

2 4647 

i®i6'23" 

2 47  00 

i°i5'26" 

2 47  13 

i°i4'3o" 

2 47  26 

1OJ3/33// 

2 47  40 

i°i2'37" 

2 47  54 

i°ii'4i" 

2 48  09 

Star. 

1909. 

1912. 

1915. 

1918. 

1921. 

1924. 

1927. 

1980. 

Polaris 

51  Cephei.. 

i°io'45'' 

2 48  24 

1°  9'49" 

2 48  39 

1°  8'53" 

2 48  55 

1“  7'58'' 

2 49  12 

1°  7'  2" 

2 49  28 

1°  6'  7" 

2 49  45 

1°  5'i2" 

2 50  01 

1°  4'i6" 

2 50  18 

To  find  the  pole  distance  for  any  intermediate  time,  make 
a linear  interpolation  between  the  two  adjacent  tabular  values. 


512 


SUR  VE  YING. 


To  observe  for  latitude  no  knowledge  of  the  geographical 
position  is  needed. 

374.  The  Observation  for  Latitude  consists  simply  in 
observing  the  altitude  of  a circumpolar  star  at  upper  or  lower 
culmination  and  correcting  this  altitude  for  the  pole  distance 
of  the  star  and  for  refraction. 

Let  0 = latitude  ; 

d = polar  distance; 
r — refraction ; 
h — altitude  ; 

then  (p  = h^^d—r\ (i) 

the  minus  sign  being  used  for  upper,  and  the  plus  sign  for 
lower,  culmination  observations.  The  value  of  r is  taken  from 
the  following  table  of  mean  refractions  computed  for  barometer 
30  inches,  and  temperature  ^0°  F. 


TABLE  OF  MEAN  REFRACTIONS. 


Altitude. 

Refraction. 

Altitude. 

Refraction. 

10° 

5'  19” 

20° 

2'  39" 

II 

4 51 

25 

2 04 

12 

4 28 

30 

I 41 

13 

4 07 

35 

I 23 

14 

3 50 

40 

I 09 

15 

3 34 

45 

0 58 

16 

3 20 

50 

0 4<9 

17 

3 08 

60 

0 34 

18 

2 58 

70 

0 21 

19 

2 48 

80 

0 10 

GEODETIC  SURVEYING. 


513 


The  index  error  of  the  vertical  circle  is  eliminated  by  read- 
ing with  the  telescope  direct  and  reversed,  providing  the  verti- 
cal circle  is  complete.  If  the  vertical  limb  is  but  an  arc  of  180° 
or  less,  the  index  error  cannot  be  eliminated  in  this  way.  In 
this  case  the  second  method  is  recommended. 

375.  First  Method. — Mount  the  instrument  firmly,  pre- 
ferably on  a post,  and  adjust  carefully  the  plate-bubbles, 
especially  the  one  parallel  to  the  plane  of  the  vertical  circle. 
About  five  or  ten  minutes  before  the  star  comes  to  its  culmi- 
nation read  the  altitude  of  the  star  with  telescope  direct. 
Revolve  the  telescope  on  its  horizontal  axis  and  also  on  its 
vertical  axis,  relevel  the  mstrument  if  the  bubbles  are  not  in  the 
middle,  but  do  not  readjust  the  bubbles,  and  bring  the  tele- 
scope upon  the  star.  Make  two  readings  in  this  position. 
Revolve  the  telescope  and  instrument  again  about  their  axes, 
relevel,  and  read  again  in  first  position.  This  gives  two  direct 
and  two  reversed  readings  taken  in  such  a way  as  to  eliminate 
the  error  from  collimation,  the  index  error  of  vertical  circle, 
and  also  the  error  of  adjustment  of  the  plate-bubbles.  The 
result,  when  corrected  for  refraction  and  the  pole  distance  of 
the  star,  should  be  the  latitude  of  the  place  within  the  limits 
of  accuracy  and  exactness  of  the  vertical  circle-readings. 

376.  Second  Method. — An  “artificial  horizon,”  formed  by 
the  free  surface  of  mercury  in  an  open  vessel,  may  be  used  in 
conjunction  either  with  the  transit  or  a sextant.  If  the  former 
is  used  two  pointings  are  made — one  to  the  star  and  the  other 
to  its  image  in  the  mercury  surface.  The  angle  measured  is 
then  twice  the  apparent  altitude  of  the  star.  The  position  of 
the  vessel  of  mercury  will  be  on  a line  as  much  below  the 
horizontal  as  the  star  is  above  it.  The  instrument  is  first  set 
up  and  then  the  artificial  horizon  put  in  place.  The  surface 
of  the  mercury  must  be  free  from  dust.  If  the  mercury  is  not 
clean  it  may  be  strained  through  a chamois-skin  or  skimmed 
by  a piece  of  cardboard.  Any  open  vessel  three  or  more 

33 


514 


SUJ^VEYING. 


inches  in  diameter  may  be  used  for  holding  the  mercury.  It 
should  be  placed  on  a solid  support  and  protected  from  the 
wind. 

The  observations  with  a transit  would  then  consist  in  taking 
a reading  on  the  star  just  before  culmination,  two  readings  on 
the  image,  and  then  one  on  the  star.  The  index  error  of  the 
vernier  on  the  vertical  circle  will  then  be  eliminated,  since  both 
plus  and  minus  angles  have  been  read,  and  their  sum  taken  for 
twice  the  altitude  of  the  star.  This  method  is  adapted  to 
transits  with  incomplete  vertical  limbs. 

The  Sextant  may  also  be  used  with  the  artificial  horizon 
and  will  give  more  accurate  results  than  can  be  obtained  with 
the  ordinary  field  transits.  The  double  altitude  angle  is  then 
measured  at  once  by  bringing  the  direct  and  reflected  images 
of  the  star  into  coincidence.  In  both  cases  the  observed  angle 
is  2^,  and  the  latitude  is  found  from  equation  (i),  as  before. 
If  there  is  much  wind  the  mercury  basin  may  be  partially 
covered,  leaving  only  a narrow  slit  in  the  vertical  plane  through 
instrument  and  star,  or  the  regular  covered  mercurial  horizon 
may  be  used.  This  is  covered  by  two  pieces  of  plate-glass  set 
at  right  angles  to  each  other  like  the  roof  of  a house.  If  the 
opposite  faces  of  these  glasses  are  not  parallel  planes,  an  error 
is  introduced.  This  is  eliminated  by  reversing  the  horizon 
apparatus  on  half  the  observations.  It  is  best,  however,  to 
avoid  the  use  of  glass  covers,  if  possible. 

If  tin-foil  be  added  to  the  mercury  an  amalgam  is  formed, 
whose  surface  remains  a perfect  mirror,  which  is  not  readily 
disturbed  by  wind.  As  much  tin-foil  should  be  used  as  the 
mercury  will  unite  with.  Observations  may  then  be  made  in 
windy  weather  without  the  aid  of  a glass  cover. 

377.  Correction  for  Observations  not  on  the  Meridian. 

If  the  star  is  more  than  five  or  ten  minutes  of  time  from  the 

meridian,  it  is  necessary  to  apply  a correction  to  the  observed 
altitude  to  give  the  altitude  at  culmination.  The  following 


GEODETIC  SURVEYING. 


515 


approximate  rule  gives  these  corrections  for  the  two  circum- 
polar stars  here  used,  with  an  error  of  less  than  i"  of  arc  when 
the  observation  is  taken  not  more  than  18  minutes  of  time 
from  the  star’s  meridian  passage,  and  the  error  is  less  than  10" 
of  arc  when  the  observation  is  made  32  minutes  of  time  from 
the  meridian. 

Rule  for  reducing  circumrneridian  altitudes  to  the  altitude  at 
culmination. 

For  Polaris : Multiply  the  square  of  the  time  from  meridian 
passage,  in  minutes,  by  0.0444,  and  the  product  is  the  correc- 
tion in  seconds  of  arc. 

For  51  Cephei : Multiply  the  square  of  time  from  meridian 
passage,  in  minutes,  by  0.1143,  and  the  product  is  the  correc- 
tion in  seconds  of  arc. 

The  correction  is  to  be  added  to  the  observed  altitude  for 
upper  culmination,  and  subtracted  for  lower  culmination. 

By  using  these  corrections  an  observation  for  latitude  may 
be  made  at  any  time  for  a period  of  about  one  hour,  near  the 
time  of  culmination. 

378.  The  Observation  for  Azimuth  is  made  on  one  of 
the  two  stars  here  chosen  when  it  is  at  or  near  its  eastern  or 
western  elongation,  for  the  same  reason  that  latitude  observa- 
tions are  taken  at  culmination.  The  azimuth  of  a star  at 
elongation  is  found  from  the  formula, 


. . , sine  of  polar  distance 

sine  of  azimuth  = ^ ^ ^ . 

cosine  01  latitude 


• (I) 


This  formula  is  so  simple  that  it  is  hardly  necessary  to  give  a 
table  of  values  of  azimuths  for  various  latitudes.  Such  a table 
is  given  for  Polaris,  however,  on  p.  33.  The  pole  distances 
are  given  on  p.  511,  and  the  latitude  is  found  by  observation. 
It  is  not  necessary  to  know  the  azimuth  of  the  star  at  elonga- 


5i6 


SURVEYING. 


tion  before  making  the  observations.  This  can  be  computed 
afterwards  from  the  observed  latitude. 

The  observation  for  azimuth  consists  simply  in  measuring 
the  horizontal  angle  between  the  star  and  some  conveniently 
located  station,  marked  by  an  artificial  light.  The  operation 
is  in  no  sense  different  from  the  measurement  of  the  horizontal 
angle  between  two  stations  at  different  elevations.  The  great 
source  of  error  is  in  the  horizontal  axis  of  the  telescope.  If 
this  is  not  truly  horizontal  then  the  line  of  sight  does  not  de- 
scribe a vertical  plane,  and  since  the  two  objects  observed  have 
very  different  elevations,  the  angle  measured  will  not  be  that 
subtended  by  vertical  planes  passing  through  the  objects  and 
the  axis  of  the  instrument.  To  eliminate  this  error  the  tele- 
scope is  reversed,  and  readings  taken  in  both  positions.  The 
following  programme  is  recommended: 


PROGRAMME  FOR  OBSERVING  FOR  AZIMUTH  ON  A CIRCUM- 
POLAR-STAR  AT  ELONGATION. 


Instrument. 

Time  of  Observation. 

Reading  on 

Direct 

lo  min.  before  elongation 

Mark. 

Reversed 

/ ••••••••• 

4 “ “ 

Star. 

ii 

2 “ “ 

<< 

Direct 

2 min.  after  “ 

<< 

4 “ “ 

ti 

7 “ “ 

Mark. 

Reversed 

lO  “ “ 

< i 

The  instrument  should  always  be  relevelled  after  reversing, 
but  the  bubbles  should  not  be  readjusted  after  the  observa- 
tions have  begun.  If  this  be  done  and  the  above  programme 
followed,  all  instrumental  errors  will  be  eliminated  except 


GEODETIC  SURVEYING. 


517 


those  of  graduation.  Of  course  both  verniers  are  to  be  read 
each  time. 

Having  found  the  latitude,  the  azimuth  of  the  star  at  elon- 
gation is  found  from  equation  (i)  above.  This  is  then  added 
to  or  subtracted  from  the  horizontal  angle  between  mark  and 
star,  as  the  case  may  be,  to  give  the  azimuth  of  the  mark  from 
the  north  point.  If  the  azimuth  is  to  be  referred  to  the  south 
point,  which  it  generally  is,  we  must  add  or  subtract  180°. 

379.  Corrections  for  Observations  near  Elongation. — 
As  in  the  case  of  observations  for  latitude,  we  may  have  an 
approximate  rule  for  reducing  an  observed  azimuth  when  near 
elongation  to  what  it  would  have  been  if  taken  at  elongation. 
The  limits  of  accuracy  are  also  about  the  same,  but  the  factors 
are  slightly  different. 

Ride  for  reducing  azimuth  observations  on  Polaris  and  51 
Cephei  near  elongation  to  their  true  values  at  elongation,  for 
latitude  40°. 

For  Polaris,  multiply  the  square  of  the  time  from  elonga- 
tion in  minutes  by  0.058,  and  the  product  will  be  the  correction 
in  seconds  of  arc. 

For  51  Cephei,  multiply  the  square  of  the  time  from  elonga- 
tion in  minutes  by  0.124,  and  the  product  will  be  the  correction 
in  seconds  of  arc. 

The  formula  for  reduction,  when  near  elongation,  is 
c — 1 12.5  f sin  i"  tan  A, 

where  c = correction  to  observed  azimuth  in  seconds  of  arc ; 
t — time  from  elongation  in  seconds  of  time; 

A = azimuth  of  star  at  elongation. 

log  1 12.5  sin  i"  = 6.7367274. 

From  this  formula  and  that  of  equation  (i)  we  may  compute 
the  coefficients  for  the  above  approximate  rules  for  any  latitude. 


518 


SURVEYING. 


Thus,  for  latitude  30°  we  have  azimuth  of  Polaris,  1885, 
3o'.4,  whence  the  coefficient  of  reduction  for  elongation  of 
Polaris  in  latitude  30°  is  found  to  beo.052,  and  for  latitude  50° 
it  is  0.069. 

For  51  Cephei,  this  coefficient  for  latitude  30°  is  o.iio,  and 
for  latitude  50°,  0.148. 

Ph'om  the  above  data  the  corrections  for  an  observation  of 
a circumpolar  star  near  elongation  may  be  computed. 

If  azimuth  be  reckoned  from  the  south  point,  as  is  common 
in  topographical  and  other  geodetic  work,  and  if  it  increase  in 
the  direction  S.W.N.E.,  then  a star  at  western  elongation  has 
an  azimuth  of  less  than  180°,  and  at  eastern  elongation  its 
azimuth  is  more  than  180°. 

The  corrections  to  reduce  to  elongation,  as  above  com- 
puted, should  be  added  to  the  computed  azimuth  of  the  star  at 
western  elongation,  and  subtracted  when  at  eastern  elongation. 

380.  The  Target. — This  may  be  a sort  of  box,  in  which  a 
light  may  be  placed.  A narrow  vertical  slit  should  be  cut,  sub- 
tending an  angle,  at  the  instrument,  from  one  to  two  seconds  of 
arc.  This  should  be  set  as  far  from  the  instrument  as  conven- 
ient, as  from  a quarter  of  a mile  to  one  mile.  The  width  of 
slit  desired  may  be  computed  for  any  given  angular  width 
and  distance  by  remembering  that  the  arc  of  one  second  is 
three-tenths  of  an  inch  for  a mile  radius.  The  target  should 
be  sufficiently  distant  to  enable  it  to  be  seen  with  the  stellar 
focus  without  appreciable  parallax,  as  the  instrument  should 
not  be  refocused  on  the  target.  This  target  may  be  set  on 
any  convenient  azimuth  from  the  observation-station,  as  upon 
one  triangulation  station  when  the  observations  are  taken  at 
another,  thus  obtaining  directly  the  azimuth  of  this  line. 

381.  Illumination  of  Cross-wires. — Various  methods  are 
used  to  illuminate  the  wires,  the  crudest  of  which  is,  perhaps,  to 
hold  a bull’s-eye  lantern  so  as  to  throw  light  down  the  tele- 
scope-tube through  the  objective,  taking  care  not  to  obstruct 
the  line  of  sight. 


GEODETIC  SURVEYING. 


519 


A very  good  reflector  may  be  made  from  a piece  of  new 
tin,  cut  and  bent  as  in  Fig.  144.  The  straight 
strip  is  bent  about  the  object  end  of  the  tele- 
scope tube,  leaving  the  annular  elliptic  piece 
projecting  over  in  front.  This  is  then  bent  to 
any  desired  angle,  preferably  about  forty-five 
degrees,  and  turned  so  that  an  attendant  can  '44- 

reflect  light  down  the  tube  by  illuminating  the  disk  from 
a convenient  position.  This  position  should  be  so  chosen 
that  the  lantern  may  throw  the  light  from  the  observer, 
rather  than  towards  him.  If  the  reflecting  side  of  the  disk  be 
whitened,  the  effect  is  very  good.  The  opening  should  be  about 
three-fourths  or  seven-eighths  inch  in  its  shorter  diameter,  the 
longer  diameter  being  such  as  to  make  its  vertical  projection 
equal  to  the  shorter  one.  There  is,  of  course,  no  necessity  of 
limiting  or  of  making  true  the  outer  edges  of  the  disk. 

TIME  AND  LONGITUDE. 

382.  Fundamental  Relations. — In  all  astronomical  compu- 
tations the  observer  is  supposed  to  be  situated  at  the  centre 
of  the  celestial  sphere  and  the  stars  appear  projected  upon  its 
surface.  Their  positions  with  respect  to  the  observer  may  be 
fixed  by  two  angular  coordinates.  The  most  common  plane  of 
reference  for  these  coordinates  is  that  of  the  celestial  equator, 
and  the  coordinates  referring  to  it  are  known  as  Right  Ascen- 
sion and  Declination — corresponding  to  Longitude  and  Lati- 
tude on  the  earth’s  surface. 

Right  ascension  is  counted  on  the  equator  from  west  tow- 
ards east.  As  a zero  of  right  ascension  the  vernal  equinox 
is  taken. 

Declination  is  counted  on  a great  circle  perpendicular  to 
the  equator,  and  is  called  positive  when  the  star  is  north  and 
negative  when  south. 


520 


SURVEYING. 


In  Fig.  145 

P is  the  pole  ; 

Z is  the  zenith  of  the  observer  ; 

5 is  the  star  ; 

Then  R.  A.  star  = VPS  — arc  VE  ; 

Dec.  star  = SE. 

These  coordinates  are  fixed,  varying  only  by  slow  changes 
due  to  the  shifting  of  the  reference-plane. 

Another  system  of  coordinates  is  often  used  in  fixing  the 
place  of  a star,  namely:  Hour-angle  and  Declination.  Hour- 
angle  is  the  angle  at  the  pole  between  the  meridian  and  the 
great  circle  passing  through  the  star  and  the  pole  perpendicu- 


z 


lar  to  the  equator.  Hour-angle  will  of  course  be  constantly 
changing  each  instant.  In  Fig.  145  hour-angle  = ZPS. 

383.  Time. — The  motion  of  the  earth  on  its  axis  is  perfect- 
ly uniform.  We  obtain,  therefore,  a uniform  measure  of  time 
by  employing  the  successive  transits  of  a point  in  the  equator 
across  the  meridian  of  any  place.  The  point  naturally  chosen 
is  the  vernal  equinox. 

A Sidereal  Day  is  the  interval  of  time  between  two  succes- 


GEODETIC  SURVEYING. 


521 


sive  upper  transits  of  the  vernal  equinox  over  the  same  merid- 
ian. 

The  Sidereal  Time  at  any  instant  is  the  hour-angle  of  the 
vernal  equinox  at  that  instant  reckoned  from  the  meridian 
westward  from  o’"  to  24*".  Thus,  when  the  vernal  equinox  is  on 
the  meridian,  the  hour-angle  is  o'"  o®  and  the  sidereal  time 
is  o™  o®.  When  the  vernal  equinox  is  west  of  the  merid- 
ian the  sidereal  time  is  o'"o®. 

We  have  in  Fig.  145 

Hour-angle  of  ver.  eq.  = ZPV  = 6 = sidereal  time; 

Right  asc.  of  star  = VPS  = a ; 

Hour-angle  of  star  = ZPS  = H; 

Whence  S — a — H, (i) 

From  this  equation,  knowing  the  sidereal  time  and  the 
R.  A.  of  the  star,  the  hour-angle  may  always  be  computed. 

When  H —o^  i.e.,  when  the  star  is  on  the  meridian,  0 = a,  or, 
in  other  words,  the  R.  A.  of  any  star  is  equal  to  the  true  local 
sidereal  time  when  the  star  is  on  the  meridian.  By  noting  the 
exact  time  of  transit  of  any  star  whose  R.  A.  is  known,  the 
local  sidereal  time  will  be  at  once  known. 

An  Apparent  Solar  Day  is  the  interval  of  time  between  two 
successive  upper  transits  of  the  true  sun  across  the  same 
meridian. 

Apparent  or  True  Solar  Time  is  the  hour-angle  of  the  true 
sun. 

Owing  to  the  annual  revolution  of  the  earth,  the  sun’s 
right  ascension  is  constantly  increasing.  It  follows,  therefore, 
that  a solar  day  is  longer  than  a sidereal  day.  In  one  year 
the  sun  moves  through  24^'^®  of  right  ascension.  There  will 
be,  therefore,  in  one  tropical  year  (which  is  the  interval  be- 


522 


SURVEYING. 


tween  two  successive  passages  of  the  sun  through  the  vernal 
equinox)  exactly  one  more  sidereal  day  than  solar  days  ; or,  in 
other  words,  in  a tropical  year  the  vernal  equinox  will  cross 
the  meridian  of  any  given  place  once  more  than  the  sun  will.^ 

The  solar  days  will,  however,  be  unequal  for  two  reasons  : 

1st.  The  sun  in  its  apparent  motion  round  the  earth  does 
not  move  in  the  equator,  but  in  the  ecliptic. 

2d.  Its  motion  in  the  ecliptic  is  not  uniform. 

On  account  of  these  inequalities  the  true  solar  day  cannot 
be  used  as  a convenient  measure  of  time.  But  a mean  solar 
day  has  been  introduced,  which  is  the  mean  of  all  the  true 
solar  days  of  the  year  and  which  is  a uniform  measure  of 
time. 

Suppose  a fictitious  sun  to  start  out  from  perigee  with  the 
true  sun,  to  move  uniformly  in  the  ecliptic,  returning  to  peri- 
gee at  the  same  moment  as  the  true  sun.  Now,  suppose  a 
second  fictitious  sun  moving  in  the  equator  in  such  a way  as 
to  make  the  circuit  of  the  equator  in  the  same  time  that  the 
first  fictitious  sun  makes  the  circuit  of  the  ecliptic,  the  two  fic- 
titious suns  starting  together  from  the  vernal  equinox  and  re- 
turning to  it  at  the  same  moment.  The  second  fictitious  sun 
will  move  uniformly  in  the  equator  and  will  be  therefore  a 
uniform  measure  of  time.  This  second  fictitious  sun  is  known 
as  the  Mean  Sun. 

A Mean  Solar  Day  is  therefore  the  interval  between  the 
upper  transits  of  the  mean  sun  over  the  meridian  of  any  place. 

Mean  Solar  Time  at  any  meridian  is  the  hour-angle  of  the 
mean  sun  at  that  meridian  counted  from  the  meridian  west 
from  to  24''^®. 

The  Equation  of  Time  is  the  quantity  to  be  added  to  or 
subtracted  from  apparent  solar  time  to  obtain  mean  time. 

The  equation  of  time  is  given  in  the  American  Ephemeris 
for  Washington  mean  and  apparent  noon  of  each  day.  If  the 
value  is  required  for  any  other  time  it  can  be  interpolated  be- 
tween tlie  values  there  given. 


GEODETIC  SURVEYING. 


523 


384.  To  convert  a Sidereal  into  a Mean-time  Interval, 
and  vice  versa. — According  to  Bessel,  the  tropical  year  con- 
tains 365.24222  mean  solar  days,  and  since  the  number  of  side- 
real days  will  be  greater  by  one  than  the  number  of  mean  solar 
days,  we  have 

365.24222  mean  sol.  days  = 366.24222  sid.  days  ; 

I mean  sol.  day  = 1.00273791  sid.  days. 

Let  /„  = mean  solar  interval ; 

Is  = sidereal  interval ; 
k = 1.00273791. 

Thus 


4 = IJ:  = /„+  IJk  - i)  = 4 + 0.00273794 ; 

Im-J  = 4 - 4 (i  - = 4 - 0.00273044. 


By  the  use  of  these  formulae  the  process  of  converting  a 
sidereal  interval  into  a mean-time  interval,  and  vice  versa,  is 
made  very  easy.  It  is  rendered  more  easy  by  the  use  of 
Tables  II.  and  III.  of  the  Appendix  to  the  American  Ephem- 
eris  and  Nautical  Almanac,  where  the  quantity  IJ^k — i)  is 


given  with  the  argument  I^,  and  4 


with  the  argument  4. 


Example. — Given  the  sidereal  interval  4 = 15^  40""  50^.50,  find 
the  corresponding  mean-time  interval. 


Is  — 15^  40™  50^.50 

Table  II.  gives  for  15*"  40"^  2 33.996 

“ 50^50  0.138 


/«.=  i5  38  16.37 


524 


SURVEYING. 


385.  To  change  Mean  Time  into  Sidereal. — Referring 
to  Fig.  145,  suppose  5 to  represent  the  mean  sun. 

Then  ZPS  = hour-angle  of  mean  sun  = mean-time  = T ; 
VPE  — R.  A.  of  mean  sun  = ; 

6 = sidereal  time. 

From  equation  (i),  p.  521, 

6 = a^-\-  T. 

The  right  ascension  of  the  mean  sun  is  given  in  the  Ameri- 
can Ephemeris  both  for  Greenwich  and  Washington  mean 
noon  of  each  date.  It  is  called  ordinarily  the  sidereal  time  of 
mean  noon,  which  is  of  course  the  right  ascension  of  the  mean 
sun  at  noon,  since  at  mean  noon  the  mean  sun  is  on  the 
meridian  and  its  right  ascension  is  equal  to  the  sidereal  time. 
Since  the  sun’s  right  ascension  increases  360°  or  24^”  in  one 
year,  it  will  change  at  the  rate  of  3"^  56®*555  in  one  day,  or 
9^.8565  in  one  hour. 

Suppose  = sid.  time  of  mean  noon  at  Greenwich; 

= “ “ “ “ “ “ the  place  for  which 

T is  known ; 

L — longitude  west  of  Greenwich. 

Then  — OJ  -4-  9^.8565^, 

where  L is  expressed  in  hours  and  decimals  of  an  hour. 
In  this  way  the  sidereal  time  of  mean  noon  may  be  obtained 
for  the  meridian  of  observation. 

Substituting  for  its  equivalent,  and  reducing  the  mean- 
time interval  to  sidereal, 

i = d,  + T+  T{k  - I). 

Example. — Longitude  of  St.  Louis,  6^  o™  49^.16  = 6^.0136. 
Mean  time,  1886,  June  10,  25™  25^.5.  Required  correspond- 

ing sidereal  time. 


GEODETIC  SURVEYING. 


52s 


From  Amer.  Ephem.,  p.  93  : 


(for  Greenwich) 
6.0136  X 9-8565 

T 


e 


T{k-  I),  Table  111.,= 


= s'"  15"  3’- 30 

= o 59-27 
= 5 16  2.57 

= 10  25  25.50 
= I 42.74 

= 15  43  10.81 


It  should  be  remarked  that  the  quantity  59^.27  will  be  a 
constant  correction,  to  be  added  to  the  sid.  time  of  mean  noon 
at  Greenwich  to  obtain  the  sid.  time  of  mean  noon  at  St. 
Louis. 

386.  To  change  from  Sidereal  to  Mean  Time. — This 
process  is  simply  the  reverse  of  that  for  changing  from  mean 
to  sidereal  time.  Using  the  same  notation  as  before,  we  shall 
have 


Subtracting  from  the  given  sidereal  time  {Q)  the  sidereal 
time  of  mean  noon  (^0),  we  have  the  sidereal  interval  elapsed 
since  mean  noon,  and  this  needs  simply  to  be  changed  into  a 
mean-time  interval. 

Example. — Given  1886,  June  10,  15^  43""  io’.8i  sidereal 
time,  to  find  the  corresponding  mean  time. 


^ = 15  43  10.81 
(as  before)  = 5 16  2.57 


— 6'^  = 10  27  8.24 
I — -i)  (Table  II.)  = i 42.74 


7^=  10  25  25.50 


526 


SURVEYING. 


387.  The  Observation  for  Time,  as  here  described,* 
consists  in  observing  the  passage,  or  transit,  of  a star  across 
the  meridian.  The  direction  of  the  meridian  is  supposed  to 
have  been  determined  by  an  azimuth  observation.  If  the  in- 
strument be  mounted  over  a station  the  azimuth  from  which 
to  some  other  visible  point  is  known,  the  telescope  can  be  put 
in  the  plane  of  the  meridian.  An  observation  of  the  passage 
of  a star  across  the  meridian  will  then  give  the  local  time,  when 
the  mean  local  time  of  trajisit  of  that  star  has  been  computed. 
In  order  to  eliminate  the  instrumental  errors  at  least  tv/o  stars 
should  be  observed,  at  about  the  same  altitude.  If  the  instru- 
ment has  no  prismatic  eye-piece,  then  only  south  stars  can  be 
observed  with  the  ordinary  field-transits;  that  is,  only  stars 
having  a south  declination,  if  the  observer  is  in  about  40°  north 
latitude.  Stars  near  the  pole  should  not  be  chosen,  since  they 
move  so  slowly  that  a small  error  in  the  instrument  would 
make  a very  large  error  in  the  time  of  passage. 

388.  Selection  of  Stars. — The  stars  should  be  chosen  in 
pairs,  each  pair  being  at  about  the  same  altitude,  or  declination. 
It  is  supposed  that  the  American  Ephemeris  is  to  be  used. 
The  “ sidereal  time  of  transit,  or  right  ascension  of  the  mean 
sun,”  is  its  angle  reckoned  easterly  on  the  equatorial  from  the 
vernal  equinox.  This  is  given  in  the  Ephemeris  for  every  day 
of  the  year.  Similarly,  the  right  ascension  of  many  fixed  stars 
is  given  for  every  ten  days  of  the  year,  under  the  head  of 

Fixed  Stars,  Apparent  Places  for  the  Upper  Transit  at  Wash- 
ington.” These  latter  change  by  a few  seconds  a year,  from 
the  fact  that  the  origin  of  coordinates,  the  vernal  equinox  itself, 
changes  by  a small  amount  annually.  If,  therefore,  the  hour- 
angle,  or  right  ascension,  of  both  the  mean  sun  and  a fixed 


* It  is  assumed  that  the  engineer  or  surveyor  has  only  the  ordinary  field- 
transit,  without  prismatic  eye-piece,  so  that  he  can  only  read  altitudes  less  than 
60°.  The  accuracy  to  be  attained  is  about  to  the  nearest  second  of  time. 


GEODETIC  SURVEYING. 


527 


star  be  found  for  any  day  of  the  year,  the  difference  will  be  the 
sidereal  interval  intervening  between  their  meridian  passages, 
the  one  having  the  greater  hour-angle  crossing  the  meridian 
much  later  than  the  other.  When  this  interval  is  changed 
to  mean  time  the  result  is  the  mean  or  clock  time  intervening 
between  their  meridian  passages.  If  a fixed  star  is  chosen 
whose  right  ascension  is  eight  hours  greater  than  that  of  the 
mean  sun  for  any  day  in  the  year,  then  this  star  will  come 
to  the  meridian  eight  hours  (sidereal  time)  after  noon,  or  at 
7^  58™  411364  after  noon  of  the  civil  day  indicated  in  the  Nau- 
tical Almanac.  If,  therefore,  one  wishes  to  make  his  observa- 
tions for  time  from  8 to  10  o’clock  P.M.  he  should  select  stars 
whose  hour-angles,  or  right  ascensions,  are  from  8 to  10  hours 
greater  than  that  of  the  mean  sun  for  the  given  date. 

In  the  following  table  such  lists  are  made  out  for  the  first  day 
of  each  month  for  the  year  1888.  The  mean  time  of  transit  is 
given  for  the  meridian  of  Washington  to  the  nearest  minute,  as 
well  as  its  mean  place  for  the  year.  None  of  these  values  will 
vary  more  than  three  or  four  minutes  from  year  to  year,  and 
therefore  the  table  may  be  used  for  any  place  and  for  any  time. 
The  table  merely  enables  the  observer  to  select  the  stars  to  be 
observed.  After  these  are  chosen  their  local  mea7i  time  of  transit 
must  be  worked  out  with  accuracy  from  the  Nautical  Almanac.* 
For  any  other  day  of  the  month  we  have  only  to  remember  that 
the  star  comes  to  the  meridian  3“  56®  earlier  (mean  time) 
each  succeeding  day,  so  that  for  n days  after  the  first  of  the 
month  we  subtract  3.93  n minutes  from  the  mean  time  of 
transit  given  in  the  table,  and  this  will  give  the  approximate 
mean  time  of  transit  for  that  date.  If  we  take  n days  before 

* Even  this  trouble  may  be  avoided  by  using  Clarke's  Transit  Tables  (Spon, 
London).  Price  to  American  purchasers  less  than  one  dollar.  They  are  pub- 
lished annually  in  advance,  and  give  the  Greenwich  mean  time  of  transit  of  the 
sun  and  many  fixed  stars  for  every  day  in  the  year.  They  are  computed  for  pop- 
ular use  from  the  Nautical  Almanac. 


LIST  OF  SOUTHERN  TIME-STARS  FOR  EACH 


528 


SU/^  VE  YING. 


LIST  OF  SOUTHERN  TIME-STARS  FOR  EACH  n—Co7ttinued. 


GEODETIC  SURVEYING. 


529 


August  i. 

Approx. 
Mean  Time 
of  Transit. 

e 

N >0  ■>r  M 00  »© 

CO  'ro  M N CO  CO  ■<r 

«)  00  0 Oi  O'  O'  c> 

October  i. 

S 

VO  ro  VO  N w 

to  0 fO  ^ N CO 

00  00  00  On  Ov 

December  i. 

B 

O'  ^ M On  fO 

Cl  to  fO  0 ''t-  to 

00  00  On  0 0 0 

c . 

^0  0 10  H o> 

M N iO  0 H M N 

00  00  00  00 

^ tx  0 VO  w VO 

CO  'tf  w N M H 

■Q  0 M H N N 

W C4  Cl  N W W 

B 

00  00  VO  tv 

HI  ro  M CO  fO 

ja 

0 0 W M W « 

.Sc 

y.S 

-«(*-  10  tn  VO  On  0 

VO  0 N 0 ‘O  W 

0 W 10  00 

N CH  fO  N W 

1 1 1 1 1 1 1 

"0  "T  3-  0 

■'T  PC  10  0 M m 

‘U  O'  N VO  00  M 

Pt  PI 

1 1 1 1 1 1 

'tv  VO  VO  fO  On  VO 

Cl  ro  to  0 

0 

O'  00  00  0 0 Cl 

1 1 1 T I + 

bD 

rt 

S . 

m to  ro  fO  CO 

-•i-  to  ^ CO  CO 

ro  M ro  ro  to 

K 

H 

C/3 

9 Ophiuchi 

b Ophiuchi 

72  Sagitt 

Sagitt 

Tj  Serpentis  

A Sagitt 

I Aquilae 

</)  Capri 

fi.  Aquarii 

C Capri 

/3  Aquarii 

B Aquarii 

y Aquarii 

• . • • • 

• ••••• 

u V a;  V u u 

u u u U u U 

0 CQ.  CO  rji  so  H 

July  i. 

Approx. 
Mean  Time 
of  Transit. 

e 

0 <0  00  00  W 0 CO 

CO  w 1-1  p)  rr  10  p) 

03  O'  O'  O'  O'  (S'  0 

Di 

U 

ca 

u 

H 

d. 

W 

(/} 

^ 10  to  0 0 VO 

0 Cl  ^ M N r4 

J3 

00  00  CO  O'  On  On 

a 

bi 

CQ 

Id 

> 

0 

12; 

B 

*-i  to  Cl  Cl  tv 

0 0 Cl  to  10  M 

CXD  00  00  00  00  O' 

w 0 0 

^ On  00  fO  M 

M to  to  0 fo  0 

10  to  VO  VO  VO 

TO  M M VO  VO  W 

M CO  to  0 M 

00  On  O'  O'  0 0 

M M M M d W 

^ M On  00  <00 

to  0 fO  0 

N S e?  c? 

rt 

•S  c 

U.2 

CJ  w 

Q 

00  00  0 H 0 to 

to  IH  CO  N H N fO 

0 

00  N On  <0  NO  0 to 

W M N M M 

1 1 1 1 1 1 1 

O'  VO  M O'  fo 

Cl  0 H 0 0 to 

0 

VO  ^ C»  w CJ 

1 1 1 1 1 1 

'%-!  ro  O'  ^ to  0 

M M ro  to  Cl 

0 

00  0 VO  oo  00  VO 

ro  H w 

1 1 1 1 1 1 

tij 

n 

S 

N N W fO  M fO  fO 

Cl  10  to  to  fO  CO 

'.t-  M Tf  to  to 

« 

H 

in 

^ Librae 

5 Scorpii 

/3*  Scorpii 

5 Ophiuchi 

a Scorpii 

^ Ophiuchi 

7}  Ophiuchi 

cr  Sagitt 

d Sagitt  

K Aquilae 

c Sagitt 

& Aquilae 

a2  Capri  

A Aquarii 

a Pis.  Aus 

(f)  Aquarii 

z'l  Aquarii 

5 Sculp 

33  Piscium 

34 


530 


SUK  VE  Y INC. 


a date  in  the  table,  add  2.93  n minutes  to  the  corresponding 
time  of  transit  to  find  the  approximate  time  of  transit  for  the 
given  date.  This  table  is  therefore  a mere  matter  of  conve- 
nience to  assist  in  selecting  the  stars  to  be  used.  They  arc 
nearly  all  southern  stars,  since  these  only  can  be  observed  with 
the  ordinary  field-transit. 

389.  Finding  the  Mean  Time  of  Transit. — As  explained 
above,  the  mean  or  clock  time  of  transit  is  simply  the  sidereal 
interval  between  the  mean  sun  and  star  for  the  given  place  and 
date,  reduced  to  mean  time.  To  find  this  interval  we  find  the 
right  ascension  of  both  mean  sun  and  star,  and  fake  their  dif- 
ference. But  the  right  ascension  or  sidereal  time  of  the  mean 
sun  or  mean  noon  is  given  for  the  meridian  of  Greenwich, 
whereas  by  the  time  the  sun  has  reached  the  given  American 
meridian  its  right  ascension  or  sidereal  time  has  increased 
somewhat,  the  hourly  increase  being  9^8565.  To  find  the 
'•sidereal  time  of  mean  noon”  for  the  given  place,  therefore, 
we  take  the  value  for  the  given  date  for  Greenwich  and  add 
to  it  9^8565  for  every  hour  of  longitude  the  place  is  west  of 
Greenwich.  This  then  gives  the  “ local  sidereal  time  of  mean 
noon.”  The  right  ascension  of  the  star,  or  the  sidereal  time  of 
its  meridian  passage,  is  then  found.  This  changes  only  by  a 
few  seconds  in  a year,  and  is  given  for  every  ten  days  in  the 
Washington  Ephemeris.  This,  therefore,  needs  no  correction 
to  reduce  it  to  its  local  value  for  any  place.  The  difference 
between  the  “ local  sidereal  time  of  mean  noon”  and  the  sidereal 
time  of  the  star  is  the  sidaral  interval  of  time  elapsing  between 
local  mean  noon  and  the  transit  of  the  star.  When  this  sidereal 
interval  is  changed  to  a mean-time  interval,  which  is  effected 
by  means  of  a table  at  the  back  of  the  Nautical  Almanac,  the 
result  is  the  local  mean  time  of  transit  of  the  star. 


GEODETIC  SURVEYING. 


531 


Example. — Compute  the  local  mean  time  of  transit  of  e Eridani  at  St.  Louis 
on  Jan.  16,  1888. 


Sidereal  time  of  mean  noon  at  Greenwich 

= 19'' 

41“ 

0 

00 

Correction  for  longitude  6.05^  west 

= + 

59  -63 

Local  sidereal  time  of  mean  noon 

= 19 

42 

27  .43 

Right  ascension  e Eridani  Jan.  16 

= 3 

27 

39  -21 

Sidereal  interval  after  mean  noon 

= 7 

45 

II  .78 

Correction  to  reduce  to  mean  time 

= — 

I 

16  .21 

Local  mean  time  of  transit 

= 7^ 

43™ 

‘ 55®.57 

390.  Finding  the  Altitude. — The  relation  between  lati- 
tude, declination,  and  altitude  is  shown  by  Fig.  146,  which  rep- 
resents a meridian  section  of  the  celestial 
sphere.  Let  PF  be  the  line  through  the 
earth’s  axis ; QQ  the  plane  of  the  equa- 
tor; Z the  zenith,  and  HH'  the  horizon. 

Then  H'P—ZQ=.  <p  is  the  latitude  of 
the  place,  and  QS  — d and  QS"  = —6" 
are  the  declinations  of  S and  S"  respec- 
tively. The  altitude  of  the  star  5 is  H'S,  Fig.  146. 

or  measured  from  tlie  south  point  it  would  be  HS,  The  alti- 
tude of  the  star  S''  is  HS" . 

We  have  therefore  for  altitude  of 


h = HZ-ZQ+  QS  = go°  - 0-fd. 

Also  for  altitude  of  S", 

h"  = HZ  - ZQ-  QS"  = 90°  - 0 - d" 

But  since  south  declination  is  considered  as  negative,  we 
have,  in  general,  for  altitude  from  the  south  point,  of  a star  in 
the  meridian, 

^ = 90°  - 0 + d. 

The  latitude  is  supposed  to  be  known  and  the  declination 


532 


SURVEYING. 


is  given  in  the  table,  whence  the  altitude  of  any  star  in  the 
list  is  readily  found. 

391.  Making  the  Observations. — The  meridian  is  sup- 
posed to  be  established.  This  may  be  done  either  by  having 
two  points  in  it  fixed,  one  of  which  is  occupied  by  the  instru- 
ment and  the  other  by  a target,  or  an  azimuth  may  be  known 
to  any  other  station  or  target.  In  either  case  the  instrument 
is  put  into  the  meridian  by  means  of  both  verniers^  either  mak- 
ing the  mean  of  the  two  read  zero  on  the  meridian  post,  or  by 
making  the  mean  of  their  readings  on  the  azimuth  station  dif- 
fer from  their  mean  reading  in  the  meridian  by  an  amount 
equal  to  the  azimuth  of  the  given  line. 

Or,  the  setting  may  be  approximately  on  the  meridian  and 
the  angle  measured  so  that  the  true  deviation  of  the  instru- 
ment from  the  meridian  is  observed  for  each  star  observation. 
The  error  in  time,  from  a given  small  error  in  azimuth,  is  then 
found  from  the  differential  equation* 


dt  = 


sin  (0  — d) 
cos  d 


da^ 


where  dt  is  the  error  in  hour-angle  in  seconds  of  arc  when  da 
is  the  deviation  from  the  meridian  in  seconds  of  arc,  0 being 
the  latitude  of  the  place,  and  S the  declination  of  the  star. 


* One  of  the  fundamental  equations  that  may  be  written  from  an  inspection 
of  Fig.  II,  p,  49,  is 

cos  5 sin  t — — cos  h sin  a, 

where  h is  the  altitude  and  /,  5^,  and  a as  above.  Differentiating  with  reference 
to  t and  <3!,  we  have 

, cos  h cos  a , 

dt= da. 

cos  o cos  t 

For  observations  very  near  the  meridian  both  cos  a and  cos  t become 
unity,  and  then  we  have 

cos  o cos  o 


GEODETIC  SURVEYING. 


533 


Having  found  the  time  correction  in  seconds  of  arc,  the 
correction  in  seconds  of  time  is  found  by  dividing  by  fifteen. 

If  the  declination  is  south,  or  negative,  the  equation  be- 
comes 


cos  O 

The  error  from  this  cause  diminishes  as  the  altitude  of  the 
star  increases,  and  is  zero  for  a zenith  observation. 

The  stars  are  chosen  in  pairs,  the  two  stars  of  a pair  hav- 
ing about  the  same  altitude  or  declination.  Thus,  from  the 
January  group  we  might  select  o‘  Eridani  and  P Orionis  as 
one  pair,  and  p Eridani  and  r Orionis  for  another.  The  stars 
are  of  course  observed  in  the  order  of  their  coming  to  the 
meridian,  irrespective  of  the  way  they  are  paired,  but  they  are 
paired  in  the  reduction. 

The  visual  angle  of  the  field  of  view  of  the  ordinary  engi- 
neer’s field-transit  is  something  over  one  degree.  The  star 
will  therefore  be  visible  in  the  telescope  for  something  over 
two  minutes  before  it  comes  to  the  vertical  wire,  it  being  here 
assumed  that  there  is  but  one  vertical  thread.  Let  an  attend- 
ant hold  the  watch  or  chronometer  and  note  the  time  to  the 
nearest  second  when  the  star  is  on  the  wire,  as  noted  by  the 
observer.  If  this  time  be  compared  with  that  of  the  computed 
mean  time  of  transit,  the  error  of  the  chronometer  is  obtained, 
so  far  as  this  observation  gives  it. 

The  instriLinent  must  be  reversed  on  the  second  star  of  each 
pair.  This  is  to  eliminate  the  instrumental  errors.  The  hori- 
zontal angle  to  the  station-mark  (whether  this  be  on  the 
meridian  or  not)  should  also  be  read  for  every  reading  on  a 
star,  or  at  least  before  and  after  the  star-readings. 

The  following  programme  would  be  adapted  to  observa- 
tions on  the  four  stars  selected  above  : 


534 


SURVEYING. 


PROGRAMME. 

1.  Set  on  azimuth  station  and  read  liorizontal  angle  (both 
verniers). 

2.  Set  in  the  meridian  and  read  both  verniers. 

3.  Set  the  approximate  altitude  of  o'  Eridani. 

4.  Note  time  of  passage  of  o'  Eridani. 

5.  Set  on  azimuth  station  and  read  both  verniers. 

6.  Set  in  the  meridian  and  read  verniers. 

7.  Note  time  of  passage  of  (i  Eridani. 

8.  Revolve  the  telescope  180°  on  its  horizontal  axis^  relevel^ 
and  read  on  the  azimuth  station. 

9.  Set  in  the  meridian  and  read  verniers. 

10.  Note  time  of  passage  of  ft  Orionis. 

11.  Note  time  of  passage  of  r Orionis. 

12.  Read  both  verniers  again  in  the  meridian  before  the 
instrument  is  disturbed. 

13.  Read  to  azimuth  station. 

We  have  thus  obtained  four  measurements  of  the  hori- 
zontal angle,  and  read  with  the  telescope  normal  and  inverted 
on  each  pair  of  stars.  Especial  care  must  be  taken  to  see  that 
the  plate-bubble  set  perpendicular  to  the  telescope  is  exactly 
in  the  centre  when  readings  are  taken  to  the  stars.  The  mean 
chronometer  error  for  the  two  stars  of  a pair  is  its  true  error, 
provided  it  has  no  rate.  If  the  chronometer  has  a known  rate, 
that  is,  if  it  is  known  to  be  gaining  or  losing  at  a certain  rate, 
then  its  error  must  be  found  for  some  particular  time,  as  that 
of  the  first  observation.  Its  rate  must  then  be  applied  to  the 
observed  time  of  transit  of  the  other  stars  for  the  intervening 
intervals  before  comparing  results.  If  local  time  alone  is  de- 
sired, the  result  is  obtained  as  soon  as  a pair  of  stars  has  been 
observed  and  their  mean  result  found. 

392.  Longitude. — If  geographical  position  or  longitude  is 
sought,  it  remains  to  compare  the  chronometer  with  the 


GEODETIC  SURVEYING. 


535 


standard  or  meridian  time  for  that  region.  This  standard  time 
is  now  transmitted  daily  from  fixed  observatories  to  almost  all 
railroad  stations  in  the  United  States.  The  time  thus  trans- 
mitted is  probably  never  in  error  more  than  a few  tenths  of  a 
second.  It  is  usually  sent  out  from  lo  A.M.  to  noon  daily.  If 
the  rate  of  the  station  clock  is  known,  and  also  that  of  the 
watch  used  in  the  time  observation,  then  a comparison  of  these 
subsequent  to  the  observation  would  give  the  difference  be- 
tween local  time  and  the  hourly  meridian  time  used,  which 
difference  changed  to  longitude  would  be  the  longitude  of  the 
place  east  or  west  of  that  standard  meridian.  If  the  station 
clock  cannot  be  relied  on  as  to  its  rate,  then  the  watch  used  in 
the  observation  must  have  a constant  known  rate.  In  this 
case  the  observer  compares  his  watch  on  the  following  day 
with  the  time  signal  as  it  is  transmitted  over  the  railroad  com- 
pany’s wires,  and  so  obtains  his  longitude. 

Local  time  can  be  observed  in  this  way  by  means  of  an  ordi- 
nary transit  to  the  nearest  second  of  time,  and  the  longitude  ob- 
tained to  the  same  accuracy  if  the  rate  of  the  chronometer  used 
is  constant  and  accurately  known.  It  is  probable,  however, 
that  several  seconds  error  may  be  made  if  a watch  is  used, 
since  probably  no  watch  has  a rate  which  is  constant  within 
one  second  in  twelve  hours.  Therefore  if  longitude  is  desired 
a portable  chronometer  should  be  used  whose  rate  is  well 
known.* 

393.  Computing  the  Geodetic  Positions. — After  the 
angles  of  the  system  are  adjusted,  and  the  sides  of  the  triangles 
computed,  we  have  the  plane  angles  and  linear  distances  from 
point  to  point  in  the  system.  It  now  remains  to  compute  the 


\ riiis  method  has  been  extensively  used  for  obtaining  approximate  geodetic 
posiiions  for  the  U.  S.  Geological  Survey  in  the  West,  comparisons  being  made 
daily  with  the  Washington  University  time  signals  which  are  transmitted  to  the 
railways  in  that  region. 


536 


SURVEYING. 


latitudes  and  longitudes  of  the  several  stations,  and  the  azi- 
muths of  the  lines. 

The  following  formulae,  though  not  exact,  arc  quite  sufYi- 
cient  when  the  sides  of  the  triangles  do  not  exceed  ten  or 
fifteen  miles  in  length  : * 


NOTATION. 


Let  L =■  latitude  of  the  known  point; 

L = latitude  of  the  unknown  point; 

M'  — longitude  of  the  known  point; 

M ■=  longitude  of  the  unknown  point ; 

Z'  = azimuth  of  the  unknown  point  from  the  known, 
counting  from  the  south  point  in  the  direction 


S.W.N.E.; 

Z = azimuth  of  the  known  point  from  the  unknown, 
or  the  back  azimuth  ; 

K = length  in  metres  of  line  joining  the  two  points; 

e = eccentricity  of  the  earth’s  meridian  section; 

N length  of  the  normal,  or  radius  of  curvature  of  a 
section  perpendicular  to  the  meridian  of  the 
middle  latitude,  in  metres. 

R ~ radius  of  curvature  of  the  meridian  in  metres. 


Then  we  have  in  terms  of  the  length  and  azimuth  of  a 
given  line,  in  seconds  of  arc,  when  the  distances  are  given  in 
metres. 


L'  -L  = -dL  BKzo^ZAr  CK^  sin^  Z+  Dh^  ^ 

M' -M= +dM=AK^^-,  [ (i) 
Z'-[-iSo°—Z  — — dZ  — dM  sin  L^n  ; J 


* For  a summarized  derivation  of  these  formulae  for  computing  the  L M 
Z’s  from  triangulation  data,  together  with  extended  tables  of  factors  used,  see 
Report  U.  S.  Coast  and  Geodetic  Survey,  1884,  Appendix  No.  7.  The  deriva- 
tion of  the  formulae  is  further  amplified  in  Appendix  D of  this  book. 


GEODETIC  SURVEYING. 


537 


where  = 


I 


n 


N2.XC1"' 

sin  L cos  L sin  i" 
— e'  sin"  L)-\ 


B = 


I 


i^arc  i 


C = 


tan  L 


= mean  latitude  = 


2RN  2.XZ  l"  ’ 

L-\-L 


and  h = value  of  first  term  in  right  member  = BK  cos  Z. 
Careful  attentioji  must  be  paid  to  the  signs  of  the  Z functions. 


TABLE  OF  LM  Z COEFFICIENTS. 


Latitude. 

Log.  A -{-  10. 

Log.  B 4-  10. 

Log.  C + 10. 

Log.  D + 10. 

30“ 

8.5093588 

8.5115729 

I . 16692 

2.3298 

31 

3363 

5054 

.18416 

•3382 

32 

3134 

4368 

.20108 

.3460 

33 

2goi 

3669 

.21772 

•3532 

34 

2665 

2959 

.23409 

•3597 

35 

8.5092425 

8.5112239 

1.25024 

2.3656 

36 

2182 

1510 

.26617 

•3709 

37 

1936 

0772 

.28193 

.3756 

38 

1687 

8.5110027 

.29753 

*.3797 

39 

1437 

8.5109275 

.31299 

•3833 

40 

8.5091184 

8.5108517 

1.32833 

2.3863 

41 

0930 

7755 

•34358 

.3888 

42 

0675 

6989 

•35875 

.3907 

43 

0419 

6220 

•37386 

.3921 

44 

8.5090162 

5449 

.38894 

• 3930 

45 

8 . 5089904 

8.5104677 

I . 40400 

2.3933 

46 

9647 

3905 

.41906 

.3932 

47 

9390 

3134 

•43414 

.3924 

48 

9133 

2364 

.44926 

.3912 

49 

8878 

1598 

•46443 

•3894 

50 

8.5088623 

8.5100835 

1.47968 

2.3871 

* is  to  be  evaluated  for  L. 

\ This  denominator  is  given  to  the  f power  in  the  Coast  Survey  formulae. 
The  rigid  development  used  in  Appendix  D.  shows  it  to  be  as  given  above. 
The  error,  however,  is  small. 


538 


SUR  VE  YING. 


Logarithmic  values  of  the  coefficients  A,  B,  C,  and  D are 
given  in  the  above  table  for  each  degree  of  latitude  from 
30°  to  50°.  By  the  aid  of  this  table  the  LM are  readily 
found.  These  tabular  values  are  computed  from  the  constants 
of  Clarke’s  spheroid.  In  this  we  have 

Equatorial  semidiameter  = 6 378  206  metres. 

Polar  semidiameter  = 6 356  584 

Whence  the  ratio  of  the  semidiameters  is 

293.98 

Clarke’s  value  of  tiie  metre  has  been  taken,  which  is 
I metre  = 39-37043  inches. 

The  difference  of  azimuth  of  the  two  ends  of  a line  is  due 
to  the  convergence  of  the  meridians  passing  through  its  ex- 
tremities, this  convergence,  as  seen  from  the  last  of  equations 
(i),  being  equal  to  the  difference  of  longitude  into  the  sine  of 
the  mean  latitude. 

When  the  sides  of  a system  of  triangulation  have  been 
computed,  and  the  azimuths  of  the  lines  are  desired  from  the 
several  stations,  the  successive  differences  of  latitude  and 
longitude  are  first  computed,  and  from  these  the  azimuths  of 
the  lines,  using  equations  (i).  If  the  longitude  is  unknown, 
the  longitude  of  the  first  station  may  be  assumed  without 
affecting  the  accuracy  of  the  computed  relative  positions.  The 
last  of  equations  (i)  gives  the  difference  between  the  forward 
and  back  azimuth  of  the  line  joining  the  two  stations.  This 
difference  being  applied,  with  the  proper  sign,  gives  the  azi- 
muth of  the  first  station  as  seen  from  the  second.  But  when 
the  azimuth  of  one  line  from  a station  is  known,  the  azimuths 
of  all  other  lines  from  that  station  are  found  from  the  adjusted 
plane-angles  at  that  station,  provided  the  spherical  excess  had 


GEODETIC  SURVEYING. 


539 


been  deducted  or  allowed  for,  in  the  adjustment.  If  no  ac- 
count has  been  taken  of  spherical  excess,  the  error  in  azimuth 
accumulates  in  working  eastward  or  westward,  and  soon  be- 
comes appreciable. 

For  any  other  station  the  azimuth  correction  is  again  found 
for  the  line  joining  this  station  with  a station  where  azimuths 
have  been  computed,  which  when  applied  gives  the  azimuth  of 
this  line  as  taken  at  the  forward  station,  whence  the  azimuths 
of  all  the  lines  from  this  station  are  known,  and  so  on. 

394.  Example. — In  Fig.  142,  p.  495,  let  the  azimuth  of  the  line  CA,  from 
C,  be  80°;  latitude  of  Cbe  40°;  the  length  of  the  line  CD  be  25000  metres  (over 
15  miles)  ; required  the  geodetic  position  of  Z>,  and  the  azimuth  of  the  line  DC 
from  D, 

COMPUTATION  OF  L M Z. 


z 

C to  A 

80° 

00' 

00". 0 

z. 

ACD  = Cs  (see  p.  495) 

39 

48 

06  .1 

z> 

C toD ... 

II9 

48 

06  . 1 

dZ 

0 

AQ  . ^ 

*T7  • D 

180° 

z 

180 

Z?  to  C 

299 

38 

16  .6 

V 

dL 

L 

40® 

+ 

00' 

6 

00'^. 000 

41  .8j7 

C 

25000  metres. 

D 

M’ 

dM 

90® 

+ 

00' 

15 

00' \ 000 
i6  .019 

40 

06 

41  .847 

M 

90 

15 

16  .019 

ist  term  — 402". 8 53 
2d  and  3d  terms  4“  i 

— dL  — 401  .847 

■^m  = 40°  °3^  22" 

B 

K 

cos  Z' 

h 

8.5108517 

4.3979400 

9.6963560 

C 

sin*  Z 

1.32833 

8.79588 

9.87679 

D 

2.3863 

5.2103 

2.6051477 

0.00100 

I .0023 

7.5966 

0.0039s 

A 

K 

sin  Z 

cos  L (a.c.) 

dM 

8.5091156 

4.3979400 

9.9383948 

0.1164540 

dM 

sin  Zn, 

- dZ 

2.96190 

9.80857 

2.9619044 
-f-  916". 019 

2.77047 

-f  589^.48 

540 


SURVEYING. 


GEODETIC  LEVELLING. 

395.  Geodetic  Levelling  is  of  two  kinds : (A)  Trigonomet- 
rical Levelling  7i\\d  {B)  Precise  Spirit-levelling.  In  trigonomet- 
rical levelling  the  relative  elevations  of  the  triangulation-sta* 
tions  are  determined  by  reading  the  vertical  angles  between  the 
stations.  When  these  are  corrected  for  curvature  of  the  earth’s 
surface  and  for  refraction  it  enables  the  actual  difference  of 
elevation  to  be  found.  In  precise  spirit-levelling  a special  type 
of  the  ordinary  spirit  or  engineer’s  level  is  used,  and  great 
care  taken  in  the  running  of  a line  of  levels  from  the  sea-coast 
inland,  connecting  directly  or  indirectly  with  the  triangulation 
stations  and  base-lines.  Both  these  methods  will  be  described. 

{A)  TRIGONOMETRICAL  LEVELLING. 

396.  Refraction. — If  rays  of  light  passed  through  the  atmos- 
phere in  straight  lines,  then  in  trigonometrical  levelling  we  should 


have  to  correct  only  for  the  curvature  of  a level  surface  at  the 
locality.  It  is  found,  however,  that  rays  of  light  near  the  sur- 
face of  the  earth  usually  are  curved  downwards — that  is,  their 
paths  are  convex  upwards.  This  curve  is  quite  variable,  some- 
times being  actually  convex  downwards  in  some  localities.  It 


GEODETIC  SURVEYING. 


541 


has  its  greatest  curvature  about  daybreak,  diminishes  rapidly 
till  8 A.M.,  and  is  nearly  constant  from  10  A.M.  till  4 P.M.,  when 
it  begins  to  increase  again.  The  curve  may  be  considered  a 
circle  having  a variable  radius,  the  mean  value  of  which  is 
about  seven  times  the  radius  of  the  earth. 

397.  Formulae  for  Reciprocal  Observations. — In  Fig.  147 
the  dotted  curve  represents  a sea-level  surface. 

Let  H — height  of  station  ^ above  sea-level ; 

H'  = height  of  station  A above  sea-level ; 

C = angle  subtended  by  the  radii  through  A and  B ; 
Z = true  zenith  distance  of  A as  seen  from  B\ 

Z'  — true  zenith  distance  of  B as  seen  from  A ; 
d — true  altitude  of  A as  seen  from  B — 90°  — Z ; 

= true  altitude  of  B as  seen  from  A — 90°  — Z'  \ 
h = apparent  altitude  of  A as  seen  from  ^ = (J  -|-  re- 
fraction ; 

h'  — apparent  altitude  of  B as  seen  from  A = d'  re- 
fraction. 

d = distance  at  sea-level  between  A and  B ; 
r = radius  of  the  earth  ; 
m = coefficient  of  refraction. 


In  the  figure  join  the  points  A and  ^ by  a straight  line. 
This  would  be  the  line  of  sight  from  A to  ^ if  there  were  no 
refraction.  Through  A and  B draw  the  radii  meeting  at  C,  ex- 
tending them  beyond  the  surface.*  Take  the  middle  point  of 
the  line  AB,  as  H,  and  draw  HC.  Take  .^^'perpendicular  to 
HC,  and  EE  through  H and  perpendicular  to  HC.  Extend 
A A'  to  meet  a perpendicular  to  it  from  B.  Then  do  we  have 


A'C  — AC\  E'E=.AD\  and 


* In  reality  these  are  the  normals  at  A and  B,  but  will  here  be  taken  as  the 

radii. 


542 


SUR  VE  YING. 


Neither  of  these  three  relations  is  quite  exact,  because 
HC  does  not  quite  bisect  the  angle  C.  The  figure  is  greatly 
exaggerated  as  compared  to  any  possible  case  in  practice, 
for  the  angle  C would  never  be  more  than  i°  in  such  work. 
The  error  in  practice  is  inappreciable. 

From  the  geometrical  relations  shown  in  the  figure  we 
have 

H - IT  ^ A'n=z  DB  scc^ (i) 

But  since  ^7  is  never  more  than  i°,  and  usually  much  less, 
we  may  say 

H-H'  =A'B  = DB  = AD\:7inBAD..  . . (2) 

But  AD  = E'E  = distance  between  the  stations  reduced  to 
their  mean  elevation  above  sea-level  = d'  ; also 


BAD  = - Z') ; 

...  tan  i{Z  - Z') (3) 

But  since  d = distance  between  stations  at  sea-leveh.  we 
have 


or 


,,  . , H+H' 

d : d ::  r-\ — : r. 


d' 


=4 


1+ 


2 

H+H' 

2r 


. . 


(4j 


whence  we  have,  for  reciprocal  observations  at  A and  B, 


= . . (5) 


GEODETIC  SURVEYING. 


543 


or,  in  terms  of  d and  6' , 

= . . (6) 

where  attention  must  be  paid  to  the  signs  of  <5^  and  8'. 

The  effect  of  refraction  is  to  increase  <5^  and  by  equal 
amounts  (presumably),  whence  their  difference  remains  unaf- 
fected. Equations  (5)  and  (6)  are  therefore  the  true  equations 
to  use  for  reciprocal  observations  at  two  stations.  Since  the 
refraction  is  so  largely  dependent  on  the  state  of  the  atmos- 
phere, the  observations  should  be  made  simultaneously  for  the 
best  results.  This  is  seldom  practicable,  however,  and  therefore 
it  is  highly  probable  that  a material  error  is  made  in  assuming 
that  the  refraction  is  the  same  at  the  two  stations  when  the 
observations  are  made  at  different  times. 

398.  Formulae  for  Observations  at  One  Station  only. — 
If  the  vertical  angle  be  read  at  only  one  of  the  two  stations, 
then  the  refraction  becomes  a function  in  the  problem.  Since 
the  curve  of  the  refracted  ray  is  assumed  to  be  circular  (it 
probably  is  not  when  stations  have  widely  different  elevations), 
the  amount  of  angular  curvature  on  a given  line  is  directly  pro- 
portional to  the  length  of  the  line  or  to  the  angle  C.  The  dif- 
ference at  ^ ox  B between  the  directions  of  the  right  line  AB 
and  the  ray  of  light  passing  between  them  is  one  half  the 
total  angular  curvature  of  the  ray  ; that  is,  it  is  the  angle 
between  the  tangent  to  the  curved  ray  at  A and  the  cord  AB. 
The  ratio  between  this  refraction  angle  at  ^ or  B and  the 
angle  {7  is  a constant  for  any  given  refraction  curve  ; that  is, 
this  ratio  does  not  change  for  different  distances  between  sta- 
tions. This  ratio  is  called  the  coefficient  of  refraction,  and  is 

Q 

here  denoted  by  m.  The  true  angle  BAD  Is  equal  to  d' 
but  since  the  observed  altitude  is  increased  by  the  amount  of 


544 


SUK  VE  YIXG. 


the  refraction,  we  have  for  the  apparent  altitude  of  B,  as  seen 
from 

Ji'  — d'  viC\ 
r 

whence  BAD  — h'-\--^ — mC. (7) 

Using  this  value  of  the  angle  BAD  in  equation  (2),  we 
obtain 

H-H’  = d tan  (//  + ^ - mc) 

= d tan  [ti  + f - w/C)  (i  + • (8) 

where  It!  is  positive  above  and  negative  below  the  horizon. 

Equation  (8)  is  used  where  the  vertical  angle  is  read  from 
one  station  only. 

Since  the  total  angular  curvature  of  the  ray  of  light  between 
A and  B is  2mC^  and  the  curvature  of  the  earth  is  we  may 
write 

C : 27nC  \\  r'  \ r,  or  r'  = — , . . . (o) 

2m 

where  r'  is  the  radius  of  the  curve  of  the  refracted  ray. 

Since  the  curvature  of  the  ray  is  of  the  same  kind  as  that 
of  the  earth,  but  less  in  amount,  the  total  correction  for  curva- 

C C 

ture  and  refraction  is  for  an  angle  equal  to mC=  —(i--2m)* 

2 2 

Also,  since  C is  always  a small  angle,  we  may  put 

C (in  seconds  of  arc)  = — f- — j,. 

^ r sm  I 

If  the  mean  radius  is  used,  we  have,  in  feet, 

log  r = 7.32020,  and  log  sin  i"  = 4.6855749, 


GEODETIC  SURVEYING. 


545 


whence  in  seconds  of  arc  and  distance  in  feet  we  have 


or 


log  C — log  d — 2.00577 


d 

101.34 


• • (10) 


or  the  curvature  is  approximately  equal  to  for  100  feet  in 
distance. 

The  following  table  gives  computed  values  of  the  combined 
mean  corrections  for  curvature  and  refraction  for  short  dis- 
tances, either  for  horizontal  or  inclined  sights.  Both  the  dis- 
tance d and  the  correction  are  in  feet,  except  for  the  last 
column,  where  the  distance  is  given  in  miles.  For  a more  ex- 
tended table  for  long  distances,  see  page  433. 


CORRECTION  FOR  EARTH’S  CURVATURE  AND  REFRACTION. 


d 

Cn 

d 

Cn 

d 

Cn 

d 

Cn 

d 

Cn 

miles. 

1 

Cn 

300 

.002 

1300 

•035 

2300 

. 108 

3300 

223 

4300 

•379 

I 

•571 

4(X) 

.003 

1400 

.040 

2400 

.118 

3400 

•237 

4400 

•397 

2.285 

500 

.005 

1500 

.046 

2500 

. 128 

3500 

.251 

4500 

•415 

3 

5-142 

600 

.007 

1600 

.052 

2600 

•139 

3600 

.266 

4600 

•434 

4 

9.141 

700 

.010 

1700 

•059 

2700 

.149 

3700 

.281 

4700 

•453 

5 

14.282 

800 

.013 

1800 

.066 

2800 

.161 

3800 

.296 

4800 

.472 

6 

20.567 

900 

.017 

1900 

.074 

2900 

.17? 

3900 

.312 

4900 

•492 

7 

27.994 

1000 

.020 

2000 

.082 

3000 

. 184 

4000 

.328 

5000 

.512 

8 

36-563 

1100 

.025 

2100 

.090 

3100 

.197 

4100 

•345 

5100 

•533 

9 

46.275 

1200 

.030 

2200 

.099 

3200 

.210 

4200 

.362 

5200 

•554 

10 

57.130 

J 

399.  Formulae  for  an  observed  Angle  of  Depression  to 
a Sea  Horizon. — In  Fig.  148  let  A be  the  point  of  observa- 
tion and  5 the  point  on  the  sea-level  surface  where  the  tangent 
from  A falls.  Then  we  have 

H=AD=^ASt^nASD 

C 

= r tan  C tan  - (ii) 


35 


* Let  the  student  prove  this  relation. 


546 


S UK  VE  Y INC. 


Since  tlie  angle  C is  always  very  small,  wc  may  let  the  arc 

equal  its  tangent,  whence 


//=-tan’(r.  . (12) 


If  the  observed  angle  of  de- 
pression ho.  h = C — mCy 


then 

and 

or 


— m 


H — - tan 
2 


H 


<'3> 

= (-) 


where  /i  is  expressed  in  seconds  of  arc. 


Log  - tan’*  i"  = 6.39032  for  distances  in  feet. 
400.  To  find  the  Value  of  m we  have 


whence 


Z = 90°  — h-\-  mC, 
Z'  — 90°  — h'  + 'tnC ; 


Z + Z'  = 180°  + 6'=  180°  — k — k'  2mC, 


GEODETIC  SURVEYING. 


547 


where  It  and  k are  the  observed  altitudes  above  the  horizon. 
It  is  evident  that  every  pair  of  reciprocal  observations  at  two 
stations  will  give  a value  for  m.  The  mean  values  of  as 
found  from  observations  on  the  United  States  Coast  Survey 
in  New  England,  were: 


Between  primary  stations,  . . , . m — 0.071 

For  small  elevations, m 0-075 

For  a sea  horizon, m — 0.078 


On  the  New  York  State  Survey  the  value  from  137  ob.ser- 
vations  was  m — 0.073.* 

H ~\- 

In  this  work  also  the  term  — — — in  equations  (4)  to  (8) 
never  affected  the  result  by  more  than  of  its  value. 

PRECISE  SPIRIT-LEVELLING. 

401.  Precise  Levelling  differs  from  ordinary  spirit-level- 
ling both  in  the  character  of  the  instruments  used  and  in  the 
methods  of  observation  and  reduction.  It  is  differential 
levelling  over  long  lines,  the  elevations  usually  being  referred 
to  mean  sea-level.  In  order  that  the  elevations  of  inland 
points,  a thousand  miles  or  more  from  the  coast,  may  be  de- 
termined with  accuracy,  the  greatest  care  is  required  to  pre- 
vent the  accumulation  of  errors.  In  order  that  triangulation 
distances  may  be  reduced  to  sea-level,  the  elevations  of  the 
bases  at  least  must  be  found.  It  is  impossible  to  carry  eleva- 
tions accurately  from  one  triangulation-station  to  another  by 
means  of  the  vertical  angles,  on  account  of  the  great  variations 
in  the  refraction.  Barometric  determinations  of  heights  are 
also  subject  to  great  uncertainties  unless  observations  be 

* See  pages  435  and  436  for  a case  of  excessive  refraction  profitably 
utilized. 


548 


SUR  VE  YING. 


made  for  long  periods.  The  only  accurate  method  of  finding 
the  elevations  of  points  in  the  interior  above  sea-level  is  by 
first  finding  what  mean  sea-level  is  at  a given  point  by  means  of 
automatic  tide-gauge  records  for  several  years,  and  then  run- 
ning a line  of  precise  spirit-levels  from  this  gauge  inland  and 
connecting  with  the  points  whose  elevations  are  required. 
Most  European  countries  have  inaugurated  such  .systems  of 
geodetic  levelling,  this  work  being  considered  an  integral  part 
of  the  trigonometrical  survey  of  those  countries.  In  the 
United  States  this  grade  of  work  was  begun  on  the  U.  S.  Lake 
Survey  in  1875,  by  carrying  a duplicate  line  of  levels  from 
Albany,  N.  Y.,  and  connecting  with  each  of  the  Great  Lakes. 
The  Mississippi  River  Commission  has  carried  such  a line 
from  Biloxi,  on  the  Gulf  of  Mexico,  to  Savannah,  Ilk,  along 
the  Mississippi  River,  and  thence  across  to  Chicago,  connect- 
ing there  with  the  Lake  Survey  Elevations.*  The  U.  S.  Coast 
and  Geodetic  Survey  is  carrying  a line  of  precise  f levels  from 
Sandy  Hook,  N.  J.,  across  the  continent,  passing  through  St. 
Louis,  their  line  here  crossing  that  of  the  Mississippi  River 
Commission.  On  all  these  lines  permanent  bench-marks  are 
left  at  intervals  of  from  one  to  five  miles,  whose  elevations 
above  mean  sea-level  are  determined  and  published. 

402.  The  Instruments  used  in  precise  levelling  differ  in 
many  respects  from  the  ordinary  wye  levels  used  in  America. 
The  levelling  instrument  prescribed  by  the  International  Geo- 
detic Commission  held  in  Berlin  in  1864  is  shown  in  Fig.  149. 

These  instruments  are  made  by  Kern  & Co.,  of  Aarau, 
Switzerland,  and  this  illustration  is  almost  an  exact  representa- 
tion of  the  instruments  used  on  the  U.  S.  Lake  and  Mississippi 
River  Surveys.^  The  bubble  is  enclosed  in  a wooden  case 
(metal  case  in  the  cut),  and  rests  on  top  of  the  pivots  or  rings; 


* The  author  had  charge  of  about  600  miles  of  this  work, 
f Called  on  that  service  geodesic  levels. 

X This  cut  is  from  Fauth’s  Catalogue,  Washington,  D.  C. 


GEODETIC  SURVEYING, 


549 


it  is  carried  in  the  hand  when  the  instrument  is  transported. 
A mirror  is  provided  which  enables  the  observer  to  read  the 
bubble  without  moving  his  eye  from  the  eye-piece.  There  is  a 
thumb-screw  with  a very  fine  thread  under  one  wye  which  is 
used  for  the  final  levelling  of  the  telescope  when  pointed  on  the 
rod.  There  are  three  levelling-screws,  and  a circular  or  box 
level  for  convenience  in  setting.  The  telescope  bubble  is  very 


Fig.  149. 


delicate,  one  division  on  the  scale  corresponding  to  about  three 
seconds  of  arc.  The  bubble-tube  is  chambered  also,  thus  al- 
lowing the  length  of  the  bubble  to  be  adjusted  to  different 
temperatures.  The  magnifying  power  is  about  45  diameters. 
There  are  three  horizontal  wires  provided,  set  at  such  a distance 
apart  that  the  wire  interval  is  about  one  hundredth  of  the  dis- 
tance to  the  rod.  The  tripod  legs  are  covered  with  white 


550 


SURVEYING. 


cloth  to  diminish  the  disturbing  effects  of  the  sun  upon  them. 
The  level  itself  is  always  kept  in  the  shade  while  at  work. 

The  levelling-rod  is  made  in  one  piece,  three  metres  long,  of 
dry  pine,  about  four  inches  wide  on  the  face,  and  strengthened 
by  a piece  at  the  back,  making  a T -shaped  cross-section.  The 
rods  are  self-reading,  that  is,  they  are  without  targets,  and  are 
graduated  to  centimetres.  An  iron  spur  is  provided  at  bottom 
which  fits  into  a socket  in  an  iron  foot-plate.  The  end  of  the 
spur  should  be  flat  and  the  bottom  of  the  socket  turned  out  to 
a spherical  form,  convex  upwards.  A box-level  is  attached  to 
the  rod  to  enable  the  rodman  to  hold  it  vertically,  and  this  in 
turn  is  adjusted  by  means  of  a plumb-line.  Two  handles  are 
provided  for  holding  the  rod,  and  a wooden  tripod  to  be  used 
in  adjusting  the  rod-bubble.  The  decimetres  are  marked  on 
one  side  of  the  graduations  and  the  centimetres  on  the  other, 
all  figures  inverted  since  the  telescope  is  inverting. 

403.  The  Instrumental  Constants  which  must  be  accu- 
rately determined  once  for  all,  but  re-examined  each  season, 
are — 

1.  The  angular  value  of  one  division  on  the  bubble-tube. 

2.  The  inequality  in  the  size  of  the  pivot-rings. 

3.  The  angular  value  of  the  wire-interval,  or  the  ratio  of 
the  intercepted  portion  on  the  rod  to  the  distance  of  the  rod 
from  the  instrument. 

4.  The  absolute  lengths  of  the  levelling-rods. 

These  constants  may  be  determined  as  follows: 

The  value  of  one  division  of  the  bubble  may  be  readily 
found  by  sighting  the  telescope  on  the  rod,  which  is  set  at  a 
known  distance  from  the  instrument,  and  running  the  bubble 
from  end  to  end  of  its  tube,  taking  rod-readings  for  each  posi- 
tion of  the  bubble.  The  bubble-graduations  are  supposed  to 
be  numbered  from  the  centre  towards  the  ends. 


GEODETIC  SURVEYING. 


551 


Let  = mean  of  all  the  eye-end  readings  of  the  bubble 
when  it  was  run  to  the  eye-end  of  its  tube  ; 

^2  = same  for  bubble  at  object-end  of  tube  ; 

= mean  of  all  the  object-end  readings  when  bubble 
was  at  eye-end  of  tube  ; 

(^2  = same  for  bubble  at  object-end  of  tube; 

= mean  reading  of  rod  for  bubble  at  eye-end , 
i^2  ==  same  for  bubble  at  object-end  ; 

D — distance  from  instrument  to  rod ; 

V — value  of  one  division  of  the  bubble  (sine  of  the 
angle)  at  a unit’s  distance. 


Then 


V = 


D 


R.-R. 

'E,  - O,  E, 


(0 


In  seconds  of  arc  we  would  have 


V (in  seconds)  = 


sin  I' 


E,  -O,  E,~  O, 


V ■ (2) 


If  a table  is  to  be  prepared  for  corrections  to  the  rod-read- 
ings for  various  distances  and  deviations  of  the  bubble  from 
the  centre  of  its  tube,  then  the  value  as  given  by  equation  (i) 
is  most  convenient  to  use.  The  value  of  one  division  of  a level 
bubble  should  be  constant,  but  it  is  often  affected  by  its  rigid 
fastenings,  which  change  their  form  from  changes  in  tempera- 
ture. 

The  inequality  in  the  size  of  the  rings  is  found  by  revers- 
ing the  bubble  on  the  rings,  and  also  reversing  the  telescope 
in  the  wyes.  The  bubble  is  reversed  only  in  order  to  eliminate 
its  error  of  adjustment.  The  following  will  illustrate  the 
method  of  making  and  reducing  the  observations: 


552 


SURVEYING. 


BunnLR-RRADINCS. 


Tel.  eye-end  north. 

Lev.  direct. 

North. 

4.3 

South. 

5-5 

it  ti  it 

“ reversed. 

4-7 

5-2 

Tel.  eye-end  south 

Lev.  direct. 

9.0 

0.2 

(-1.7) 

— 0.42 

10.7 

3.7 

it  it  it 

“ reversed. 

6.6 

3-3 

Tel.  eye-end  north 

Lev.  direc^. 

12.8 

4.4 

(+5.8) 

+ I -45 

7-0 

5-5 

“ “ south 

“ reversed. 

5-2 

Mean  reading  north 
“ “ south 

North  minus  south 

= — 0.40 
= + 1-45 
= - 1.85 

9.2 

(-1.5) 

— 0.38 

10  7 

That  is  to  say,  the  bubble  moves  1.85  divisions  towards  the 
object-cud  when  the  telescope  is  reversed  in  the  wyes.  This  is 
evidently  twice  the  inequality  of  the  pivot-rings  ; and  since  the 
axis  of  a cone  is  inclined  to  one  of  its  elements  by  one  half 
the  angle  at  the  apex,  so  the  line  of  sight  is  inclined  to  the 
tops  of  the  rings  by  one  fourth  of  1.85  divisions,  or  0.46  divi- 
sions of  the  bubble.  It  is  also  evident  that  the  eye-end  ring 
is  the  smaller,  and  that  therefore  when  the  top  surfaces  of  the 
rings  are  horizontal  the  line  of  sight  inclines  downward  from 
the  instrument.  The  correction  is  therefore  positive.  This  is 
called  the  pivot-correction^  and  changes  only  with  an  unequal 
wear  in  the  pivot-rings. 

The  angidar  value  of  the  wire-interval  is  found  by  measur- 
ing a base  on  level  ground  of  about  300  feet  from  an  initial 
point  -f-/*  in  front  of  the  objective.  Focus  the  telescope 
on  a very  distant  object,  and  measure  the  distance  from  the 
inside  of  the  objective  to  the  cross-wires,  this  being  the  value 

* See  art.  205  for  the  significance  of  these  terms,  as  well  as  for  the  theory 
of  the  problem. 


GEODETIC  SURVEYING. 


553 


of  y for  that  instrument.  Measure  the  space  intercepted  on 
the  rod  between  the  extreme  cross-wires. 


\{  d—  length  of  base,  counting  from  the  initial  point ; 
s = length  of  the  intercepted  portion  of  the  rod ; 

r ='C  = constant  ratio  of  distance  to  intercept; 
then  r = - ; 


and  for  any  other  intercept  s'  on  the  rod  we  have 

d'  = rs'  +/+C (3) 

When  r, /,  and  c are  found,  a table  can  be  prepared  giving 
distances  in  terms  of  the  wire-intervals. 

T/ie  errors  m the  absolute  lengths  of  the  rods  affect  only 
the  final  differences  of  elevation  between  bench-marks.  This 
correction  is  usually  inappreciable  for  moderate  heights. 

404.  The  Daily  Adjustments. — The  adjustments  which 
are  examined  at  the  beginning  and  close  of  each  day’s  work 
are  as  follows : 

1.  The  collimation,  that  is,  the  amount  by  which  the  line 
of  sight,  as  determined  by  the  mean  reading  of  the  three  wires, 
deviates  from  the  line  joining  the  centres  of  the  rings. 

2.  The  bubble-adjustment — that  is,  the  inclination  of  the 
axis  of  the  bubble  to  the  top  surface  of  the  rings. 

3.  The  rod-level.  This  is  examined  only  at  the  beginning 
of  each  day’s  work,  and  made  sufficiently  perfect. 

The  first  two  adjustments  are  very  important,  since  it  is  by 
means  of  these  (in  conjunction  with  the  pivot-correction, 
determined  once  for  the  season)  that  the  relation  of  the  bubble 


554 


SUA'  VE  Y INC. 


to  the  line  of  siglit  is  found.  It  is  not  customary  in  this  work 
to  try  to  reduce  these  errors  to  zero,  but  to  make  them  reason- 
ably small,  and  then  determine  iJieir  values  and  correct  for 
them.  It  is  evident  that  if  the  back  and  fore  sights  be  kept 
exactly  equal  between  bench-marks,  then  the  errors  in  the 
instrumental  adjustments  are  fully  eliminated  ; and  in  any  case 
these  errors  can  only  affect  the  excess  in  length  of  the  sum  of 
the  one  over  that  of  the  other.  It  is  to  this  excess  in  length 
of  back-sights  over  fore-sights,  or  vice  versa,  that  the  instru- 
mental constants  are  applied  ; but  in  order  to  apply  them  their 
values  must  be  accurately  determined.  The  curvature  of  a 
level  surface  would  also  enter  into  this  excess,  but  it  is  usually 
so  sm.all  a residual  distance,  that  the  correction  for  curvature 
is  quite  insignificant.  There  are,  however,  three  instrumental 
corrections  to  be  applied  for  the  amount  of  the  excess,  namely, 
the  corrections  for  collimation,  inclination  of  bubble,  and  in- 
equality of  pivots,  designated  respectively  by  e,  i,  and  p.  Since 
three  horizontal  wires  are  read  on  the  rod,  the  wire-intervals 
can  be  used  in  place  of  the  distances,  for  they  are  linear  func- 
tions practically,  and  so  a record  is  kept  of  the  continued  sum 
of  the  lengths  of  the  back  and  fore  sights,  and  from  these  the 
final  difference  is  found. 

The  colliniation-correction  is  taken  out  for  a distance  of 
one  unit  (the  metre  has  been  universally  used  in  this  kind  of 
levelling),  and  then  the  correction  for  any  given  case  found  by 
multiplying  by  the  residual  distance. 

Let  = rod-reading  for  telescope  normal ; 

“ inverted ; 

d — distance  of  rod  from  instrument. 

2d 


Then 


(I) 


GEODETIC  SURVEYING. 


555 


« The  correction  for  the  inclination  of  the  bubble  to  the  tops 
of  the  rings  is  found  by  reversing  the  bubble  on  the  telescope 
and  reading  it  in  both  positions.  In  such  observations  the 
initial  and  final  readings  are  taken  with  the  bubble  in  the  same 
position,  thus  giving  an  odd  number  of  observations.  Usually 
two  direct  and  one  reversed  reading  are  taken.  The  correction 
is  found  in  terms  of  divisions  on  the  bubble,  the  correction  in 
elevation  being  taken  from  the  table  prepared  for  that  purpose. 


Let  — mean  of  the  eye-end*  readings  for  level  direct ; 


E,  = “ “ 

a 

a 

“ “ reversed ; 

0,  = “ “ 

object 

(( 

“ “ direct  ; 

0,  = “ “ 

u 

(( 

“ “ reversed ; 

then 


(2) 


The  pivot  correction  has  already  been  found,  and  is  sup* 
posed  to  remain  constant  for  the  season. 

If  E be  the  excess  of  the  sum  of  the  back-sights  over  that 
of  the  fore-sights,  then  the  final  correction  for  this  excess  is 


(3) 

where  v is  taken  from  eq.  (i),  p.  551.  Evidently,  if  the  fore- 
sights are  in  excess,  the  correction  is  of  the  opposite  sign. 

405.  Field  Methods. — The  great  accuracy  attained  in  pre- 
cise levelling  is  due  quite  as  much  to  the  methods  used  and 
precautions  taken  in  making  the  observations  as  to  the  instru- 
mental means  employed.  Aside  from  errors  of  observation 
and  instrumental  errors,  we  have  two  other  general  classes  of 


* By  eye-end  is  always  meant  the  end  towards  the  eye-end  of  the  telescope, 
whether  in  a direct  or  a reversed  position. 


556 


SURVEYING. 


errors,  which  can  be  avoided  only  by  proper  care  being  used 
in  doing  tlie  work.  Tliese  two  classes  arc  errors  from  unstable 
supports  and  atmospheric  errors. 

Any  settling  ot  the  rod  between  the  fore  and  back  readings 
upon  it  will  result  in  the  final  elevation  being  too  high,  while 
any  settling  of  the  instrument  between  the  back  and  fore 
readings  from  it  will  also  result  in  too  high  a final  elevation. 
Such  errors  are  therefore  cumulative,  and  the  only  way  in 
which  they  can  be  eliminated  is  to  duplicate  the  work  over 
the  same  ground  in  the  opposite  dircctio7i.  As  a general  pre- 
caution, the  duplicate  line  should  always  be  run  in  the  opposite 
direction.  This  will  result  in  larger  discrepaficies  than  if  both 
are  run  in  the  same  direction,  but  the  mean  is  nearer  the  truth. 

Atmospheric  errors  may  come  from  wind,  heated  air-cur- 
rents causing  the  object  sighted  to  tremble  or  “dance,”  or 
from  variable  refraction.'  For  moderate  winds  the  instrument 
may  be  shielded  by  a screen  or  tent,  but  if  its  velocity  is  more 
than  eight  or  ten  miles  an  hour,  work  must  be  abandoned. 
To  avoid  the  evil  effects  of  an  unsteady  atmosphere  the  length 
of  the  sights  is  shortened  ; but  when  a reading  cannot  be  well 
taken  at  a distance  of  about  150  feet,  or  50  metres,  it  would 
be  better  to  stop,  since  the  errors  arising  from  the  number  of 
stations  occupied  would  make  the  work  poor.  At  about  8 
o’clock  A.M.  and  4 P.M.  very  large  changes  in  the  refraction 
have  been  observed  on  lines  over  ground  which  is  passing  from 
sun  to  shade,  or  vice  versa,  when  the  image  was  apparently 
very  steady.  In  clear  weather  not  more  than  three  or  four 
hours  a day  can  be  utilized  for  the  best  work,  and  sometimes, 
with  hot  days  and  cool  nights,  it  is  impossible  to  get  an  hour 
when  good  work  can  be  done. 

In  making  the  observations  the  bubble  is  brought  exactly 
to  the  centre  of  its  tube,  the  observer  being  able  to  do  this 
by  means  of  the  thumb-screw  under  one  wye,  and  the  mirror 
which  reflects  the  image  of  the  bubble  to  the  observer  at  the 


GEODETIC  SURVEYING. 


557 


eye-piece.  If  there  is  no  mirror  to  the  bubble,  then  it  is 
brought  approximately  to  the  centre,  and  the  recorder  reads 
it  while  the  observer  is  reading  the  three  horizontal  wires.  In 
any  case  the  bubble-reading  is  recorded  in  the  note-book,  and 
if  it  was  not  in  the  middle  a correction  is  made  for  the  eccen- 
tric position  by  means  of  a table  prepared  for  the  purpose. 
The  mean  of  the  three  wire-readings  is  taken  as  the  reading 
of  that  rod,  the  observer  estimating  the  tenths  of  the  centi- 
metre spaces,  thus  reading  each  wire  to  the  nearest  millimetre. 
The  wires  should  be  about  equally  spaced  so  that  the  mean  of 
the  three  wires  coincides  very  nearly  with  the  middle  wire. 
The  differences  between  the  middle  and  extreme  wire-readings 
are  also  taken  out  to  give  the  distance,  as  well  as  to  check  the 
readings  themselves  by  noting  the  relation  of  the  two  intervals. 
If  they  are  not  about  equal,  then  one  or  more  of  the  three 
readings  is  erroneous.  This  is  a most  important  check,  and 
constitutes  an  essential  feature  of  the  method. 

It  has  been  found  economical  to  have  two  rodmen  to  each 
instrument,  so  that  no  time  shall  be  lost  between  the  back  and 
fore  sight  readings  from  an  instrument-station.  Since  but  a 
small  portion  of  the  day  can  generally  be  utilized,  it  is  desira- 
ble to  make  very  rapid  progress  when  the  weather  is  favora- 
ble. When  two  rodmen  are  used,  and  the  air  is  so  steady  that 
lOO-metre  sights  can  be  taken,*  it  is  not  difficult  for  an  expe- 
rienced observer  to  move  at  the  rate  of  a mile  an  hour. 

On  the  U.  S.  Coast  and  Geodetic  Survey  a much  more 
laborious  method  of  observing  than  the  one  above  outlined 
has  been  followed.  There  a special  kind  of  target-rod  has 
been  employed,  the  target  being  set  approximately  and 
clamped.  The  thumb-screw  under  the  wye  is  used  as  a mi- 
crometer-screw, and  two  readings  are  taken  on  it  one  when 


* This  is  about  the  limiting  length  of  sight  for  first-class  work,  even  under 
the  most  favorable  conditions. 


558 


SURVEYING. 


the  bubble  is  in  the  middle  and  the  other  when  the  centre 
wire  bisects  the  target,  the  bubble  now  not  being  in  the 
middle,  since  the  target’s  position  was  only  approximate.  The 
bubble  is  then  reversed,  and  two  more  readings  of  the  screw 
taken.  The  telescope  is  now  revolved  in  the  wyes,  and  read- 
ings taken  again  with  bubble  direct  and  reversed.  Thus  there 
are  four  independent  readuigs  taken  on  the  rod,  each  necessi- 
tating two  micrometer-readings.  The  reduction  is  also  very 
complicated,  each  sight  being  corrected  for  curvature  and  re- 
fraction  as  well  as  for  instrumental  constants.  The  duplicate 
line  is  carried  along  with  the  first  one  by  having  two  sets  of 
turning-points  for  each  instrument-station.  The  instrument, 
however,  is  set  but  once,  so  that  the  lines  are  not  wholly  inde- 
pendent. The  alternate  sections  are  run  in  opposite  directions, 
thus  partly  obviating  the  objection  to  running  both  lines  in 
the  same  direction.  The  method  first  described  was  used  on 
the  U.  S.  Lake  and  Mississippi  River  surveys,  and  is  also  the 
method  used  on  most  of  the  European  surveys  of  this  char- 
acter. 

The  instrument  is  always  shaded  from  the  sun,  both  while 
standing  and  while  being  carried  between  stations.  It  is  abso- 
lutely necessary  to  do  this  in  order  to  keep  the  adjustments 
approximately  constant,  and  the  bubble  from  continually 
moving. 

406.  Limits  of  Error. — On  the  U.  S.  Coast  and  Geodetic 
Survey  the  limit  of  discrepancy  between  duplicate  lines  is 
5mm  f 2K  where  K is  the  distance  in  kilometres.  On  the  U.  S. 
Lake  Survey  the  limit  was  10"^™  and  on  the  Mississippi 
River  Survey  it  was  f K.  These  limits  are  respectively 

0.029  0.041,  and  0.021  feet  into  the  square  root  of  the  distance 
in  miles.  If  any  discrepancies  occurred  greater  than  these  the 
stretch  had  to  be  run  again. 

The  probable  error”  of  the  mean  of  several  observations 
on  the  same  quantity  is  a function  of  the  discrepancies  of  the 


GEODETIC  SURVEYING. 


559 


several  results  from  the  mean.  If  etc.,  be  the  several 

residuals  obtained  by  subtracting  the  several  results  from  the 
mean,  and  if  ^[vv]  be  the  sum  of  the  squares  of  these  residu- 
als, and  if  m be  the  number  of  observations,  then  the  probable 


error  of  the  mean  is  7?  = ± -6745 


'^\yv\ 
m{m  — i) 


This  is  the  function  which  is  universally  adopted  for  meas- 
uring the  relative  accuracy  of  different  sets  of  observations. 

If  there  be  but  two  observations  this  formula  reduces  to 


R^±W. 

where  V is  the  discrepancy  between  two  results. 

The  European  International  Geodetic  Association  have 
fixed  on  the  following  limits  of  probable  error  per  kilometre 
in  the  mean  or  adopted  result:  4;  3““  per  km.  is  tolerable; 
± per  km.  is  too  large ; ± 2"^“  per  km.  is  fair ; and'  d:  i™"' 
per  km.  is  a very  high  degree  of  pre'cision.  On  the  U.  S. 
coast  and  geodetic  line  from  Sandy  Hook  to  St.  Louis,  a dis- 
tance of  1009  rniles,  the  probable  error  per  kilometre  was 
dh  1.2™"".*  For  the  670  miles  of  this  work  on  the  Mississippi 
River  Survey,  of  which  the  author  had  charge,  the  probable 
error  of  the  mean  for  the  entire  distance  was  23.5™”"  (less  than 
one  inch),  and  the  probable  error  per  kilometre  was  ± 0.7“™.t 
Of  course  very  little  can  be  predicated  on  these  results  as  to  the 
actual  errors  of  the  work,  since  the  number  of  observations  on 
each  value  was  usually  but  two  ; but  they  may  fairly  be  used 
for  the  purpose  of  comparing  the  relative  accuracy  of  different 
lines  where  this  function  has  been  computed  from  similar 
data. 

407.  Adjustment  of  Polygonal  Systems  in  Levelling. — If 


* Report  U.  S,  Coast  and  Geodetic  Survey,  1882,  p.  522. 
f Reports  of  the  Miss.  Riv.  Commission  for  the  years  1882,  1883,  and  1884. 


560 


SU/^  VE  Y I NG. 


a line  of  levels  closes  upon  itself  the  summation  of  all  the  differ- 
ences of  elevation  between  successive  benches  should  be  zero. 
If  it  is  not,  the  residual  error  must  be  distributed  among  the 
several  sides,  or  stretches,  composing  the  polygon,  according 
to  some  law,  so  that  the  final  corrections  which  arc  applied  to 
the  several  sides  shall  be  independent  of  all  personal  considera- 
tions. These  corrections  should  also  be  the  most  probable 
corrections.  There  are  two  general  criterions  on  which  to 
found  a theory  of  probabilities.  One  may  be  called  a prioriy 
and  the  other  a posteriori.  By  the  former  we  would  say  that 
the  errors  made  are  some  function  of  the  distance  run,  as  that 
they  are  directly  proportional  to  this  distance,  or  to  the  square 
root  of  this  distance,  etc.;  while  by  the  latter,  or  a posteriori 
method,  we  would  say  the  errors  made  on  the  several  lines  are 
a function  of  the  discrepancies  found  between  the  duplicate 
measurements  on  those  lines,  or  to  the  computed  “ probable 
error  per  kilometre,”  as  found  from  these  discrepancies.  Both 
methods  are  largely  used  in  the  adjustment  of  observations. 
These  laws  of  distribution  are  equivalent  to  establishing  a 
method  of  weighting  the  several  sides  of  the  system,  a larger 
weight  implying  that  a larger  share  of  the  total  error  is  to  be 
given  to  that  side.  When  any  system  of  weights  is  fixed  upon, 
then  the  corrections  may  be  computed  by  the  methods  of  least 
squares  so  as  to  comply  with  the  condition  that  the  corrections 
shall  be  the  most  probable  ones  for  that  system  of  weighting. 
The  most  probable  set  of  corrections  is  that  set  the  sum  of 
whose  squares  is  a minimum.  If  the  system  includes  more 
than  a few  polygons,  this  method  of  reduction  is  exceedingly 
laborious,  while  the  increased  accuracy  is  very  small  over  that 
from  a much  simpler  method. 

Fig.  150  represents  the  Bavarian  network  of  geodetic  levels, 
there  being  four  polygons.  Every  side  has  been  levelled,  and 
the  difference  of  elevation  of  its  extremities  found.  These  ele- 
vations must  now  be  adjusted  so  that  the  differences  of  eleva- 


GEODETIC  SURVEYING. 


561 


tion  on  each  polygon  shall  sum  up  zero.  When  these  sums 
are  taken  the  following  residuals  are  found  : L, -|- 20.2““  ; II., 
+ 39-3“"';  III.,  — 25.2"'"' ; and 
IV.,  -f-  108. It  was  sup-  ^ 


posed  that  an  error  of  one  deci- 
metre had  been  made  in  the 
fourth  polygon,  but  in  the  ab- 
sence of  any  knowledge  in  the 
case  this  error  must  be  distrib- 
uted with  the  rest. 


The  method  which  the  au-  q( 


thor  would  recommend  is  a 


modification  of  Bauernfeind’s,  ^ ,, 

' riG.  150. 

in  that  the  errors  are  to  be  made 

proportional  to  the  square  roots  of  the  lengths  of  the  sides  in- 
stead of  the  lengths  of  the  sides  directly.  Since  the  errors  in 
levelling  are  compensating  in  their  nature  they  would  be  ex- 
pected to  increase  with  the  square  root  of  the  length  of  the 
line,  and  it  is  the  author’s  experience  that  the  error  is  much 
nearer  proportional  to  the  square  root  of  the  distance  than  to 
the  distance  itself. 

Instead  of  treating  the  four  polygons  as  one  system  and 
solving  by  least  squares,  the  polygon  having  the  largest  error 
of  closure  is  first  adjusted  by  distributing  the  error  among  its 
sides  in  proportion  to  the  square  roots  of  the  lengths  of  those 
sides.  Then  the  polygon  having  the  next  largest  error  is  ad- 
justed, using  the  new  value  for  the  adjusted  side,  if  it  is  con- 
tiguous to  the  former  one,  and  distributing  the  remaining 
error  among  the  remaining  sides  of  the  figure,  leaving  the 
previously  adjusted  side  undisturbed.  The  adjustment  pro- 
ceeds in  this  manner  until  all  the  polygons  are  adjusted.  The 
Bavarian  system  is  worked  out  on  this  plan  in  the  following 
tabulated  form  : 


562 


SUR  VE  Y INC. 


ADJUSTMENT  OF  THE  BAVARIAN  SYSTEM  OF  LEVEL 
POLYGONS. 


No. 

Side. 

Length. 

Sq.  Root 
of 

Length 
= A. 

No. 

Polygon. 

2A. 

Difference 

of 

Elevation. 

Error 

of 

Closure 

Cor- 

rected 

Error 

of 

Closure 

Cor- 

rection. 

Corrected 

Difference 

of 

Elevation. 

I 

km. 

125.8 

II  .2 

I. 

24.6 

m. 

+ 35-8723 

mm. 

20.2 

+ 3»-3 

- 14.3 

+ 35-8580 

2 

179.0 

13-4 

I. 

— 217.5062 

- 17.0 

- 217.5232 

3 

147-3 

12. 1 

II. 

± 181.6541 

+ 39.3 

+ 39-3 

— II. I 

± 181.6652 

4 

60.6 

7.8 

II. 

43-1 

+ 32.0958 

- 7.1 

-f-  32.0887 

5 

174.0 

13-2 

II. 

-f-  179-5981 

— 12.0 

+ 179.5861 

6 

lOI  . I 

10. 0 

II. 

20.9 

T 30.0005 

— 25.2 

+ 19-9 

- 9-1 

T 30.0096 

7 

134-9 

II. 6 

III. 

— 38.6644 

— 11.0 

- 38.6754 

8 

80. 1 

9.0 

IV. 

T 48.8053 

— 36.0 

± 48.7693 

9 

87.0 

9-3 

III. 

+ 57-4440 

- 8.9 

+ 57-4351 

10 

96.8 

9.8 

IV. 

27.0 

— 100.1619 

108.0 

+ 108.0 

- 39.2 

— 100.2011 

II 

67.9 

8.2 

IV. 

+ 51-4646 

— 32.8 

+ 51-4318 

Beginning  with  polygon  IV.,  we  find  its  error  of  closure  to 
be  -j-  loS.o'"™,  this  being  distributed  among  the  three  sides  so 
that  goes  to  side  8,/^  to  side  10,  and  to  side  ii. 
The  corrected  values  for  these  sides  are  now  found.  Next 
take  the  polygon  having  the  next  largest  error  of  closure, 
which  is  number  II.,  and  distribute  its  error  in  like  manner. 
This  leaves  polygons  I.  and  III.  to  be  adjusted,  one  side  of 
the  former  and  two  of  the  latter  being  already  adjusted.  The 
corrected  errors  of  closure  for  these  polygons  are  31.3'"™  and 
respectively,  the  former  to  be  di.stributed  between  the 
sides  I and  2 and  the  latter  between  the  sides  7 and  9.  The 
resulting  corrected  values  cause  all  the  polygons  to  sum  up 
zero. 

The  sum  of  the  squares  of  the  corrections  here  found  is 
50.02  square  centimetres,  whereas  if  the  differences  of  eleva- 
tion had  been  weighted  in  proportion  to  the  lengths  of  the 
sides  and  the  system  adjusted  rigidly  by  least  squares  the  sum 
of  the  squares  of  the  corrections  would  have  been  49.65  square 
centimetres,  showing  that  the  method  here  used  is  practically 


GEODETIC  SURVEYING. 


563 


as  good  as  the  rigid  method  which  is  commonly  used.  It  has 
been  found  in  practice  to  give,  in  general,  about  the  same 
sized  corrections  as  the  rigid  system. 

408.  Determination  of  the  Elevation  of  Mean  Tide. — 
To  determine  accurately  the  elevation  of  mean  tide  at  any 
point  on  the  coast  requires  continuous  observations  by  means 
of  an  automatic  self-registering  gauge  for  a period  of  several 
years.  The  methods  of  making  these  observations  with  cuts 
of  the  instruments  employed  are  given  in  Appendix  No.  8 of 
the  U.  S.  Coast  Survey  Report  for  1876.  A float,  inclosed  in 
a perforated  box,  rises  and  falls  with  the  tide,  and  this  motion, 
properly  reduced  in  scale  by  appropriate  mechanism,  is  re- 
corded by  a pencil  on  a continuous  roll  of  paper  which  is  moved 
over  a drum  at  a uniform  rate  by  means  of  clockwork.  An 
outer  staff-gauge  is  read  one  or  more  times  a day  by  the  at- 
tendant, who  records  the  height  of  the  water  and  the  time  of 
the  observation  on  the  continuous  roll.  This  outer  staff  is 
connected  with  fixed  bench-marks  in  the  locality  by  very 
careful  levelling,  and  this  connection  is  repeated  at  intervals  to 
test  the  stability  of  the  gauge. 

To  find  from  this  automatic  record  the  height  of  mean  tide, 
ordinates  are  measured  from  the  datum-line  of  the  sheet  to 
the  tide-curve  for  each  hour  of  the  day  throughout  the  entire 
period.  This  period  should  be  a certain  number  of  entire 
lunar  months.  The  mean  of  all  the  hourly  readings  for  the 
period  maybe  taken  as  mean  tide.  It  maybe  found  advisable 
to  reject  all  readings  in  stormy  weather,  in  which  case  the 
entire  lunation  should  be  rejected. 


CHAPTER  XV. 


PROJECTION  OF  MAPS,  MAP-LETTERING,  AND  TOPO- 
GRAPHICAL SYMBOLS. 

I.  PROJECTION  OF  MAPS. 

409.  The  particular  method  that  should  be  employed  in 
representing  portions  of  the  earth’s  surface  on  a plane  sheet 
or  map  depends,  yfrj/,  on  the  extent  of  the  region  to  be  repre- 
sented ; second,  on  the  use  to  be  made  of  the  map  or  chart ; 
and  third,  on  the  degree  of  accuracy  desired. 

Thus,  a given  kind  of  projection  may  suffice  for  a small 
region,  but  the  approximation  may  become  too  inaccurate 
when  extended  over  a large  area.  It  is  quite  impossible  to 
represent  a spherical  surface  on  a plane  without  sacrificing 
something  in  the  accuracy  of  the  relative  distances,  courses, 
or  areas  ; and  the  use  to  which  the  chart  is  to  be  put  must  de- 
termine which  of  these  conditions  should  be  fulfilled  at  the 
expense  of  the  others.  A great  many  methods  have  been 
proposed  and  used  for  accomplishing  various  ends,  some  of 
which  will  be  described. 

410.  Rectangular  Projection. — In  this  method  the  merid- 
ians are  all  drawn  as  straight  parallel  lines ; and  the  parallels 
are  also  straight,  and  at  right  angles  with  the  meridians.  A 
central  meridian  is  drawn,  and  divided  into  minutes  of  latitude 
according  to  the  value  of  these  at  that  latitude  as  given  in 
Table  XI.  Through  these  points  of  division  draw  the  paral- 
lels of  latitude  as  right  lines  perpendicular  to  the  central 
meridian.  On  the  central  parallel  lay  off  the  minutes  of 


PROJECTION  OF  MAPS. 


565 


longitude,  according  to  their  value  for  the  given  latitude,  by 
Table  XL;  and  through  these  points  of  division  draw  the  other 
meridians  parallel  with  the  first. 

The  largest  error  here  is  in  assuming  the  meridians  to  be 
parallel.  For  the  latitude  of  40°,  two  meridians  a mile  apart 
will  converge  at  the  rate  of  about  a foot  per  mile.  A knowl- 
edge of  this  fact  will  enable  the  draughtsman  to  decide  when 
this  method  is  sufficiently  accurate  for  his  purpose.  Thus,  for 
an  area  of  ten  miles  square,  the  distortion  at  the  extreme  cor- 
ners in  longitude,  with  reference  to  the  centre  of  the  map  as 
an  origin  of  coordinates,  will  be  about  twenty-five  feet.  At 
the  equator  this  method  is  strictly  correct. 

In  this  kind  of  projection,  whether  plotted  from  polar  or 
rectangular  coordinates,  or  from  latitudes  and  longitudes,  all 
straight  lines  of  the  survey,  whether  determined  by  triangula- 
tion or  run  out  by  a transit  on  the  ground,  will  be  straight  on 
the  map  ; that  is,  the  fore  and  back  azimuth  of  a line  is  the 
same,  or,  in  other  words,  a straight  line  on  the  drawing  gives 
a constant  angle  with  all  the  meridians. 

This  is  the  method  to  use  on  field-sheets,  where  the  survey 
has  all  been  referred  to  a single  meridian. 

411.  Trapezoidal  Projection. — Here  the  meridians  are 
made  to  converge  properly,  but  both  they  and  the  parallels 
are  straight  lines.  A central  meridian  is  first  drawn,  and  grad- 
uated to  degrees  or  minutes ; and  through  these  points  paral- 
lels are  drawn,  as  before.  Two  of  these  parallels  are  selected  ; 
one  about  one  fourth  the  height  of  the  map  from  the  bottom, 
and  the  other  the  same  distance  from  the  top.  These  paral- 
lels are  then  subdivided,  according  to  their  respective  lati- 
tudes, from  Table  XI. ; and  through  the  corresponding  points 
of  division  the  remaining  meridians  are  drawn  as  straight  lines. 
The  map  is  thus  divided  into  a series  of  trapezoids.  The 
parallels  are  perpendicular  to  but  one  of  the  meridians.  The 
principal  distortion  comes  from  the  parallels  being  drawn  as 


566 


SUR  VE  YING, 


straight  lines,  and  amounts  to  about  thirty-two  feet  in  ten 
miles  in  latitude  40°,  and  is  nearly  proportional  to  the  square 
of  the  distance  east  or  west  from  the  central  meridian. 

The  work  should  be  plotted  from  computed  latitudes  and 
longitudes.  The  method  is  adapted  to  a scheme  which  has  a 
system  of  triangulation  for  its  basis,  the  geodetic  position  of 
the  stations  having  been  determined.  These  conditions  would 
be  fulfilled  in  a State  topographical  or  geological  survey  for 
the  separate  sheets,  each  sheet  covering  an  area  of  not  more 
than  twenty-five  miles  square. 

412.  The  Simple  Conic  .P/oiection.— In  this  projection, 
points  on  a spherical  surface  are  first  projected  upon  the  sur- 
face of  a tangent  cone,  and  then  this  conical  surface  is  devel- 
oped into  the  plane  of  the  map.  The  apex  of  the  cone  is 
taken  in  the  extended  axis  of  the  earth,  at  such  an  altitude 
that  the  cone  becomes  tangent  to  the  earth’s  surface  at  the 
middle  parallel  of  the  map.  When  this  conical  surface  is  de- 
veloped into  a plane,  the  meridians  are  straight  lines  converg- 
ing to  the  apex  of  the  cone,  and  the  parallels  are  arcs  of  con- 
centric circles  about  the  apex  as  the  common  centre. 

The  sheet  is  laid  out  as  follows:  Draw  a central  meridian, 
and  graduate  it  to  degrees  or  minutes,  according  to  their  true 
values  as  given  in  Table  XI.  Through  these  points  of  divi- 
sion draw  parallel  circular  arcs,  using  the  apex  of  the  cone  as 
the  common  centre.  For  values  of  the  length  of  the  side  of 
the  tangent  cone,  which  is  the  length  of  the  central  parallel 
above,  see  Table  XI.  The  rectangular  coordinates  of  points 
in  these  curves  are  also  given  in  the  same  table. 

On  the  middle  parallel  of  the  map  the  degrees  or  minutes 
of  longitude  are  laid  off,  and  through  these  are  drawn  the  re- 
maining meridians  as  straight  lines  radiating  from  the  apex 
of  the  tangent  cone. 

It  will  be  seen  that  the  latitudes  are  correctly  laid  off,  and 
the  degrees  of  longitude  will  be  sufficiently  accurate  for  a map 


PROJECTION  OF  MAPS. 


567 


covering  an  area  of  several  hundred  miles  square.  The  merid- 
ians and  parallels  are  at  right  angles. 

In  this  projection  the  degrees  of  longitude  on  all  parallels, 
except  the  middle  one,  are  too  great ; and  therefore  the  area 
represented  on  the  map  is  greater  than  the  corresponding  area 
on  the  sphere. 

The  chart  should  be  plotted  from  computed  latitudes  and 
longitudes. 

413.  De  ITsle’s  Conic  Projection. — This  is  very  similar 
to  the  above,  except  that  two  parallels,  one  at  one  fourth,  and 
one  at  three  fourths  the  height  of  the  map,  are  properly  grad- 
uated, and  the  meridians  drawn  as  straight  lines  through  these 
points  of  division.  The  parallels  are  drawn  as  concentric  cir- 
cles, as  in  the  simple  conic  projection.  This  is  therefore  but  a 
combination  of  the  second  and  third  methods,  and  is  more 
accurate  than  either  of  them.  The  cone  here  is  no  longer  tan- 
gent, but  intersects  the  sphere  in  the  two  graduated  parallels. 
In  this  case  the  region  between  the  parallels  of  intersection  is 
shown  too  small,  and  that  outside  these  lines  is  shown  too 
large  ; so  that  the  area  of  the  whole  map  will  correspond  very 
closely  to  the  corresponding  area  on  the  sphere.  When  these 
parallels  are  so  selected  that  these  areas  will  be  to  each  other 
exactly  as  the  scale  of  the  drawing,  then  it  is  called  “ Mur- 
doch’s projection.” 

414.  Bonne’s  Projection. — This  differs  from  the  simple 
conic  in  this — that  all  the  parallels  are  properly  graduated, 
and  the  meridians  drawn  to  connect  the  corresponding  points 
of  division  in  the  parallels.  These  latter  are,  however,  still 
concentric  circles.  The  meridians  are  at  right  angles  to  the 
parallels  only  in  the  middle  portion  of  the  map.  The  same 
scale  applies  to  all  parts  of  the  chart.  There  is  a slight  dis- 
tortion at  the  extreme  corners,  from  the  parallels  being  arcs 
of  concentric  circles.  The  proportionate  equality  of  areas  is 


568 


SU/^VEVING. 


preserved.  A rhumb-line  appears  as  a curve ; but  when  once 
drawn,  its  length  may  be  properly  scaled. 

It  will  be  noted  that  the  last  three  methods  involve  the 
use  of  but  one  tangent  or  intersecting  cone. 

415.  The  Polyconic  Projection. — For  very  large  areas  it 
is  preferable  to  have  each  parallel  the  development  of  the 
base  of  a cone  tangent  in  the  plane  of  the  given  parallel. 
This  projection  differs  from  Bonne’s  only  in  the  fact  that  the 
parallels  are  no  longer  concentric  arcs,  but  each  is  drawn  with 
a radius  equal  to  the  side  of  the  cone  which  is  tangent  at 
that  latitude.  These,  of  course,  decrease  towards  the  pole ; 
and  therefore  the  parallels  diverge  from  each  other  towards 
the  edge  of  the  chart.  The  result  of  this  is,  that  a degree 
of  latitude  at  the  side  of  the  map  is  not  equal  to  a degree 
on  the  central  meridian  ; or,  in  other  words,  the  same  scale 
cannot  be  applied  to  all  parts  of  the  map.  These  defects  ap- 
pear, however,  only  on  maps  representing  very  large  areas. 
The  whole  of  North  America  could  be  represented  by  this 
method  without  any  material  distortion. 

This  method  of  projection  is  exclusively  used  on  the  Unit- 
ed States  Coast  and  Geodetic  Survey,  and  for  all  other  maps 
and  charts  of  large  areas  in  this  country.  Extensive  tables  are 
published  by  the  War  and  Navy  Departments,  and  also  by 
the  Coast  Survey,  to  facilitate  the  projection  of  maps  by  the 
polyconic  system.  Table  VIII.  gives  in  a condensed  form  the 
rectangular  coordinates  of  the  points  of  intersection  of  the 
parallels  and  meridians  referred  to  the  intersection  of  the  sev- 
eral parallels  with  the  central  meridian  as  the  several  origins. 

416.  Formulae  used  in  the  Projection  of  Maps.* — The 
fundamental  relations  on  which  the  method  of  polyconic  pro- 
jection rests  are  given  in  the  following  formulae : 


* See  Appendix  D for  the  derivation  of  equations  (i)  and  (2). 


PROJECTION  OF  MAPS. 


569 


Normal,  being  the  radius  of  curvature 

of  a section  perpendicular  to  the  ^ 

meridian  at  a given  point N = 7 (i) 

^ ^ (l  — rsm*Z)^’ 

where  Re  is  the  equational  radius, 

e is  the  eccentricity, 

and  L is  the  latitude. 


Radius  of  the  meridian 

/p  _ pj-k}  f) 

(2) 

Radius  of  the  parallel 

(3) 

Degree  of  the  meridian 

• • • 

(4) 

=:  36ooZ;«  sin  i'\ 

Degree  of  the  parallel 

= IIS'"-  • • • 

(5) 

= 3600^^  sin 

Radius  of  the  developed  parallel,  or 

side  of  tangent  cone r = iVcot  Z.  . . . (6) 

If  n be  any  arc  of  a parallel,  in  degrees,  or  any  difference 
of  longitude  from  the  central  meridian  of*  the  drawing,  and 
if  6 be  the  corresponding  angle,  in  degrees,  at  the  vertex  of 
the  tangent  cone,  subtended  by  the  developed  parallel,  then 
since  the  angular  value  of  arcs  of  given  lengths  are  inversely 
as  their  radii,  we  have 


6 


n 


Rp 


sin  L, 


or 


6 = n sin  L, 


(8) 


570 


SURVEYING. 


Since  the  developed  parallels  are  circular  arcs,  the  rectangu- 
lar coordinates  of  any  point  an  angular  distance  of  d from 
the  central  meridian  is, 


Meridian  distance,  d^n  — x = r sin  <9.  "j 

Divergence  of  parallels,  dp  = y — r vers  6.  V.  . (9) 

= X tan  ) 


For  arcs  of  small  extent,  the  parallel  may  be  considered 
coincident  with  its  chord ; but  the  angle  between  the  axis  of  x 
and  the  chord  is  If,  then,  the  length  of  the  arc,  which  is 
7iDpj  be  represented  by  the  chord,  we  may  write 

d^  — meridian  distance  x = iiDp  cos  ^ 

and  dp  = divergence  of  parallels  = y = nDp  sin  ^6.  j 


If,  now,  dtn-,  = meridian  distance  for  i degree  of  longitude, 
and  d„ift  = meridian  distance  for  71  degrees  of  longitude. 


we  have 


d,j^  _ 7iDp  cos  \en 
dm\  Dp  cos 


But  6 n sm  Z,  so  that  = 1°  X sin  L = 38'  for  latitude  40^ 
Therefore 


cos  = cos  19'  = I,  nearly; 


PROJECTION  OF  MAPS. 


571 


For  L = 30°,  we  have  sin  L = Therefore,  for  latitude  30°, 
= n cos  \n=  n cos  (0.25;^),  nearly. 

If  we  have  obtained  the  meridian  distance,  for  i degree 
of  longitude,  and  wish  to  obtain  it  for  n degrees  in  latitude 
30°,  we  have  but  to  multiply  the  distance  for  i degree  by  n 
cos  (0.2  5 ?2). 

417.  In  Table  VIII.  the  meridian  distances  are  given,  at  vari- 
ous latitudes,  for  a difference  of  longitude  of  one  degree.  To 
find  the  meridian  distance  for  an}^  number  of  degrees  or  parts 
of  degrees,  multiply  the  distance  for  one  degree  by  the  factor 
there  given  for  the  given  latitude.  The  factor  given  in  the 
table  for  latitude  30°  is  n cos  (0.288/2),  in  place  of  71  cos  (0.25/2), 
as  obtained  above.  The  difference  is  a correction  which  has 
been  introduced  to  compensate  the  error ‘made  in  assuming 
that  the  chord  was  equal  in  length  to  its  arc.  The  corrected 
factors  enable  the  table  to  be  used  without  material  error  up 
to  25  degrees  longitude  either  side  of  the  central  meridian. 

To  obtain  the  divergence  of  the  parallels  for  differences  of 
longitude  more  or  less  than  one  degree,  multiply  the  diver- 
gence for  one  degree  by  the  square  of  the  number  of  degrees. 
It  is  evident  that  this  rule  is  based  on  the  assumption  that  the 
arc  of  the  developed  parallel  is  a parabola,  and  so  it  may  be 
considered  for  a distance  of  25  degrees  either  side  of  the  cen- 
tral meridian  between  the  latitudes  30°  and  50°  without  mate- 
rial error. 

If  the  whole  of  the  United  States  were  projected  by  this 
table,  using  the  factors  given,  to  a scale  of  i to  1,500,000,  thus 
giving  a map  some  8 by  10  feet,  the  maximum  deviation  of 
the  meridians  and  parallels  from  their  true  positions  (which 
would  be  at  the  upper  corners)  would  be  but  about  0.02  inch. 


572 


S UK  VE  Y/iVG. 


Thus,  for  a map  of  this  size,  covering  20  degrees  of  lati- 
tude and  50  degrees  of  longitude,  the  geodetic  lines  would 

have  their  true  position  within  the 
width  of  a fine  pencil  line,  by  the  use 
of  Table  VIII.  Fig.  151  will  illus- 
trate the  use  of  the  table  in  project- 
ing a map  by  the  polyconic  method. 
The  map  covers  30  degrees  in  lati- 
tude (30°  to  60°)  and  60  degrees  in 
^ longitude.  The  straight  line  0^0^  is 

first  drawn  through  the  centre  of  the  map,  and  graduated  ac- 
cording to  the  lengths  of  one  degree  of  latitude,  as  given  in 
the  second  column  of  Table  VIII.  Through  these  points  of  di- 
vision the  lines  in\  are  drawn  in  pencil  at  right  angles  to 
the  central  meridian.  On  these  lines  the  points  etc., 

are  laid  off  by  the  aid  of  the  first  part  of  Table  VIII.  This  ta- 
ble gives  the  meridian  distances  when  n is  less  than  one  degree, 
as  well  as  when  71  is  greater.  From  the  points  7nx^  777^,  etc.^ 
the  divergence  of  the  parallels  is  laid  off  above  the  lines 
by  the  aid  of  the  second  portion  of  Table  VIII.,  thus  obtaining 
the  positions  of  the  points  etc.  The  points  p mark  the 

intersection  of  the  meridians  and  parallels  ; and  these  may 
be  drawn  as  straight  lines  between  these  points,  provided  a 
sufficient  number  of  such  points  have  been  located.  The  map 
is  then  to  be  plotted  upon  the  chart  from  computed  latitudes 
and  longitudes. 


418.  Summary. — We  have  seen  that  there  are,  in  general, 
two  ways  of  plotting  a map  or  chart,  and  two  corresponding 
uses  to  which  it  may  be  put: 

First.  We  may  plot  by  a system  of  plane  coordinates, 
either  polar  (azimuth  and  distance)  or  rectangular  (latitudes 
and  departures).  This  gives  a map  from  which  distance, 
azimuth  (referred  to  the  meridian  of  the  map),  and  areas  are 
correctly  determined. 


PROJECTION  OF  MAPS. 


573 


Second.  We  may  plot  the  map  by  computed  latitudes  and 
longitudes,  and  determine  from  it  the  relative  position  of  points 
in  terms  of  their  latitude  and  longitude. 

The  first  system  is  adapted  to  small  field  sheets  and  detail 
charts  for  which  the  notes  were  taken  by  referring  all  points 
to  a single  point  and  meridian.  For  this  purpose  the  system 
of  rectangular  projection  should  be  selected,  as  long  as  the 
area  of  the  chart  is  not  more  than  about  one  hundred  square 
miles.  If  it  be  larger  than  this,  the  trapezoidal  system  should 
be  used.  In  case  this  is  done,  the  work  is  still  plotted  as 
before,  provided  it  has  all  been  referred  to  a given  meridian  in 
the  field  work,  and  then  converging  meridians  are  drawn  as 
described  above.  From  such  a chart,  not  only  the  azimuth 
(referred  to  the  central  meridian)  and  distance  may  be  deter- 
mined, but  the  correct  longitude  and  nearly  correct  latitude 
are  given. 

In  the  case  of  topographical  charts,  based  on  a system  of 
triangulation,  each  sheet  is  referred  to  a meridian  passing 
through  a triangulation-station  on  that  sheet,  or  near  to  it, 
and  the  rectangular  system  used. 

In  the  case  of  a survey  of  a long  and  narrow  belt,  as 
for  a river,  railroad,  or  canal,  if  the  survey  was  based  on  a 
system  of  triangulation,  the  convergence  of  meridians  has  been 
looked  after  in  the  computation  of  the  geodetic  positions  of 
these  stations,  and  each  sheet  is  plotted  by  the  rectangular 
system,  being  referred  to  the  meridian  through  the  adjacent 
triangulation-station.  When  many  of  these  are  combined  into 
a single  map  on  a small  scale,  then  they  must  be  plotted  on 
the  condensed  map  by  latitudes  and  longitudes,  these  being 
taken  from  the  small  rectangular  projections,  and  plotted  on 
the  reduced  chart  in  polyconic  projection  ; the  meridians  and 
parallels  having  been  laid  out  as  shown  above. 

In  case  the  belt  extends  mostly  east  and  west,  and  is  not 
based  on  a triangulation  scheme,  then  observations  for  azimuth 


574 


SU/i!  VE  Y I NG. 


should  be  made  as  often  as  every  fifty  miles.  It  will  not  do 
to  run  on  a given  azimuth  for  this  distance,  however;  for  there 
has  been  a change  in  the  direction  of  the  parallel  (or  meridian) 
in  this  distance,  in  latitude  40°,  of  about  40  minutes.  Accord- 
ing to  the  accuracy  with  which  the  Avork  is  done,  therefore, 
when  running  wholly  by  back  azimuths,  the  setting  of  the  in- 
strument must  be  changed.  Thus,  if  in  going  i degree  (53 
miles),  cast  or  west,  in  latitude  40°,  the  meridian  has  shifted 
40',  then  in  going  13  miles  cast  or  west  the  meridian  has 
changed  10';  and  this  is  surely  a sufficiently  large  correction 
to  make  it  worth  while  to  apply  it. 

When  running  west,  this  correction  is  applied  in  the  direc- 
tion of  the  hands  of  a watch,  and  when  running  east,  in  the 
opposite  direction;  that  is,  having  run  west  13  miles  by  back 
azimuth,  then  the  pointing  which  appears  north  is  really  10' 
west  of  north,  and  the  telescope  must  be  shifted  10'  around  to 
the  right. 

If  the  azimuth  be  corrected  in  this  way,  a survey  can  be 
carried  by  back  azimuth  an  indefinite  distance,  and  still  have 
the  entire  survey  referred  to  the  true  meridian. 

419.  The  Angle  of  Convergence  of  Meridians  is  the 
angle  6 in  the  equations  given  in  the  above  formula.  Then 

6 = 11  sin  Z,* 

where  n is  the  angular  change  in  degrees  of  longitude,  and  L 
is  the  latitude  of  the  place. 

For  Z = 30°,  sin  or,  in  latitude  30°  a change  of 

longitude  of  one  degree  changes  the  direction  of  the  meridian 
by  30  minutes. 

For  Z = 40°,  sin  Z = 0.643  ; or,  a change  of  longitude  of 
one  degree  changes  the  direction  of  the  meridian  by  0.643  of 
60  minutes,  or  38.6  minutes,  being  approximately  40  minutes. 

For  Z = 50°,  sin  L—  o.y66\  or,  in  going  east  or  west  one 


* From  Eq.  (G),  p.  621,  when  cos  \ A L\s  taken  as  unity. 


MAP.LETTERING  AND  TOPOGRAPHICAL  SYMBOLS.  575 


degree,  the  meridian  changes  0.766  X 60  minutes  ==  46  min- 
utes, or  approximately  50  minutes. 

Therefore  we  may  have  the  approximate  rule,  that  a change 
of  longitude  of  one  degree  changes  the  azimuth  by  as  many 
minutes  as  equals  the  degrees  of  latitude  of  the  place.  This 
rule  gives  results  very  near  the  truth  between  plus  and  minus 
40°  latitude,  that  is,  over  an  equatorial  belt  80  degrees  in 
width. 

II.  MAP-LETTERING  AND  TOPOGRAPHICAL  SYMBOLS. 

420.  Map-Lettering. — The  best-drawn  map  may  have  its 
appearance  ruined  by  the  poor  skill  or  bad  taste  displayed  in 
the  lettering.  The  letters  should  be  simple,  neat,  and  dignified 
in  appearance,  and  have  their  size  properly  proportioned  to  the 
subject.  The  map  is  lettered  before  the  topographical  symbols 
are  drawn.  When  a map  is  drawn  for  popular  display,  some 
ornamentation  may  be  allowed  in  the  title  ; but  even  then, 
the  lettering  on  the  map  itself  should  be  plain  and  simple. 
When  the  map  is  for  official  or  professional  use,  even  the  title 
should  be  made  plain. 

On  Plate  IV.  are  given  several  sets  of  alphabets  which  are 
well  adapted  to  map  work.  Of  course  the  size  should  vary 
according  to  the  scale  of  the  map  and  the  subject,  as  shown  on 
Plate  V.  It  is  a good  rule  to  make  all  words  connected  with 
water  in  italics.  The  small  letters  in  stump  writing  will  be 
found  very  useful,  and  these  should  be  practised  thorougjily. 
The  italic  capitals  go  with  these  small  letters  also. 

In  place  of  the  system  of  letters  above  described,  and 
which  has  heretofore  been  almost  exclusively  used  for  map- 
ping purposes,  a new  system,  called  “ round  writing,”  may  be 
used.  A text-book  on  this  method,  by  F.  Soennecken,  is  pub- 
lished by  Messrs.  Kueffel  & Esser,  New  York.  This  work  is 
done  with  blunt  pens,  all  lines  being  made  with  a single  stroke. 


576 


SU/^VEV/NG. 


It  looks  well  when  well  done,  and  requires  but  a small  fraction 
of  the  time  required  to  make  the  ordinary  letters,  h'or  work- 
ing drawings  and  field  maps  it  is  especially  adapted. 

421.  Topographical  Symbols, — In  topographical  repre- 
sentation, where  elevations  have  been  taken  sufficiently  num- 
erous and  accurate,  the  outline  of  the  ground  should  be  rep- 
resented by  contours  rather  than  by  hachurcs,  or  hill  shading, 
which  simply  gives  an  approximate  notion  of  the  slope  of  the 
ground,  but  no  indication  of  its  actual  elevation.  Where  the 
ground  has  so  steep  a slope  that  the  contour  lines  would  fall 
one  upon  another,  it  is  well  here  to  put  in  shading-lines,  as 
shown  on  Plate  III.  The  water  surfaces  and  streams  may  be 
water-lined  in  blue,  or  left  white.  The  contour  lines  over  al- 
luvial ground  should  be  in  brown  (crimson  and  burnt  sienna), 
while  those  over  rocky  and  barren  ground  should  be  in  black. 
This  is  a very  simple  and  effective  method  of  showing  the 
character  of  the  soil. 

The  practices  of  the  government  surveys  should  be  fol- 
lowed in  the  matter  of  conventional  surface  representation, 
such  as  meadow,  swamp,  woodland,  prairie,  cane-brake,  etc., 
with  all  their  varieties.  Some  of  these  are  given  in  the  United 
States  Coast  Survey  Report  for  1879  and  1883,  while  Plate  III. 
shows  most  of  those  used  on  the  Mississippi  River  Survey. 
Those  shown  in  Plate  II.  are  adapted  to  higher  latitudes,  and 
are  those  used  in  the  field-practice  surveys  at  Washington 
University.  This  plate  is  an  exact  copy  of  one  of  the  annual 
maps  made  from  actual  surveys  by  the  Sophomore  class.  On 
these  the  contours  are  all  in  black,  for  the  purpose  of  photo- 
lithographing. 


PLATE  1. 


Ining  space  each  year,  except  on  the  Pacific  coast,  where 


ISOCONIC  CHART  FOR  1885. 


Reduced  from  U.  S.  Coast  ard  Geodetic 


PLATE  1 


litu(le_  Vj&st-  from  j Greenu/ich. 


fOVAl.  / 


'?VENn\ 


LincoiIn 


ICOCOY 


ENVER 


LOUIS 


^es^oni 


SCALE  OF  STATUTE  mI 


RAn\mC/^ALLY  a CO.,  iA/GR’S,  CHICAGO. 


NOTE.-AII  isogonic  lines  are  moving  towards  the  left  westerly) 


, at  an  average  rate  of  one-twentieth  ^1-20)  the  intervening  space  each  year,  except  on  the  Pacific  coast,  where 
there  is  a very  slow  movement  in. the  opposite  direction. 


1 

' — 1 

1 


TOPOGRAPHICAF.  PRACTICE  SURVEY 


1886 


SW  EET  SPRINGS  MO. 

by  the 

CLASS 


POL Y 1’E(  H N 1 C SCIK ) OL 

of 

WASl  1 lNGr(  )N  I'N  f VERSI T Y 


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ConveiitioTial  Si^iis  Tor 

T 01^0  GRAPHICAL  MAPS, 

SCAI>E  1:  10000  . 

T) e signed  for  Photo -Ptho^aphin^. 


PLATE  m. 


:«iva»utKi;iiDa(foit 


Pvavna  and  engrave ^ bj 

EDWARD  MOLITOR,  T.E. 


APPENDICES. 


APPENDIX  A. 


THE  JUDICIAL  FUNCTIONS  OF  SURVEYORS. 

BY  JUSTICE  COOLEY  OF  THE  MICHIGAN  SUPREME  COURT. 

When  a man  has  had  a training  in  one  of  the  exact  sciences,  where 
every  problem  within  its  purview  is  supposed  to  be  susceptible  of  accu- 
rate solution,  he  is  likely  to  be  not  a little  impatient  when  he  is  told  that, 
under  some  circumstances,  he  must  recognize  inaccuracies,  and  govern 
his  action  by  facts  which  lead  him  away  from  the  results  which  theoreti- 
cally he  ought  to  reach.  Observation  warrants  us  in  saying  that  this  re- 
mark may  frequently  be  made  of  surveyors. 

In  the  State  of  Michigan  all  our  lands  are  supposed  to  have  been 
surveyed  once  or  more,  and  permanent  monum.ents  fixed  to  determine 
the  boundaries  of  those  who  should  become  proprietors.  The  United 
States,  as  original  owner,  caused  them  all  to  be  surveyed  once  by  sworn 
officers,  and  as  the  plan  of  subdivision  was  simple,  and  was  uniform  over 
a large  extent  of  territory,  there  should  have  been,  with  due  care,  few  or 
no  mistakes;  and  long  rows  of  monuments  should  have  been  perfect 
guides  to  the  place  of  any  one  that  chanced  to  be  missing.  The  truth 
unfortunately  is  that  the  lines  were  very  carelessly  run,  the  monuments 
inaccurately  placed ; and,  as  the  recorded  witnesses  to  these  were  many 
times  wanting  in  permanency,  it  is  often  the  case  that  when  the  monument 
was  not  correctly  placed  it  is  impossible  to  determine  by  the  record,  with 
the  aid  of  anything  on  the  ground,  where  it  was  located.  The  incorrect 
record  of  course  becomes  worse  than  useless  when  the  witnesses  it  refers 
to  have  disappeared. 

It  is,  perhaps,  generally  supposed  that  our  town  plats  were  more  ac- 
curately surveyed,  as  indeed  they  should  have  been,  for  in  general  there 
can  have  been  no  difficulty  in  making  them  sufficiently  perfect  for  all 
practical  purposes.  Many  of  them,  however,  were  laid  out  in  the  woods; 
some  of  them  by  proprietors  themselves,  without  either  chain  or  com- 
pass, and  some  by  imperfectly  trained  surveyors,  who,  when  land  was 
cheap,  did  not  appreciate  the  importance  of  having  correct  lines  to  deter- 
mine boundaries  when  land  should  become  dear.  The  fact  probably  is 
that  town  surveys  are  quite  as  inaccurate  as  those  made  under  authority 
of  the  general  government. 

It  is  now  upwards  of  fifty  years  since  a major  part  of  the  public  sur- 
veys in  what  is  now  the  State  of  Michigan  were  made  under  authority  of 


58o 


SUN  VE  YING. 


the  United  States.  Of  tlie  lands  south  of  Lansing,  it  is  now  forty  years 
since  the  major  part  were  sold  and  the  work  of  improvement  begun.  A 
generation  has  passed  away  since  tliey  were  converted  into  cultivated 
farms,  and  few  if  any  of  the  original  corner  and  quarter  stakes  now  re- 
main. 

The  corner  and  quarter  stakes  were  often  nothing  but  green  sticks 
driven  into  the  ground.  Stones  might  be  put  around  or  over  these  if 
they  were  handy,  but  often  they  were  not,  and  the  witness  trees  must  be 
relied  upon  after  the  stake  was  gone.  Too  often  tlic  first  settlers  were 
careless  in  fi.xing  their  lines  with  accuracy  while  monunuMits  remained, 
and  an  irregular  brush  fence,  or  something  equally  unirustwortliy,  may 
have  been  relied  upon  to  keep  in  mind  where  the  blazed  line  once  was. 
A fire  running  through  this  might  sweep  it  away,  and  if  nothing  were  sub- 
stituted in  its  j)lace,  the  adjoining  proprietors  might  in  a few  years  be 
found  disputing  over  their  lines,  and  perhaps  rushing  into  litigation,  as 
soon  as  they  had  occasion  to  cultivate  the  land  along  the  boundary. 

If  now  the  disputing  parties  call  in  a surveyor,  it  is  not  likely  that  any 
one  summoned  would  doubt  or  question  that  his  duty  was  to  find,  if 
possible,  the  place  of  the  original  stakes  which  determined  the  boundary 
line  between  the  proprietors.  However  erroneous  may  have  been  the 
original  survey,  the  monuments  that  were  set  must  nevertheless  govern, 
even  though  the  effect  be  to  make  one  half-quarter* section  ninety  acres 
and  the  one  adjoining  but  seventy;  for  parties  buy  or  are  supposed  to 
buy  in  reference  to  those  monuments,  and  are  entitled  to  what  is  within 
their  lines,  and  no  more,  be  it  more  or  less.  Mclver  ?/.  lVa//cer,4  Whea- 
ton’s Reports,  444;  Land  Co.  v.  Satinders,  103  U.  S.  Reports,  316;  Cot- 
tingha7n  v.  Parr,  93  111.  Reports.  233;  Bimtoji  v.  Cardwell,  53  Texas  Re- 
ports, 408:  lVaiso7i  V.  Jones,  85  Penn.  Reports,  117. 

While  the  witness  trees  remain  there  can  generally  be  no  difficulty  in 
determining  the  locality  of  the  stakes.  When  the  witness  trees  are 
gone,  so  that  there  is  no  longer  record  evidence  of  the  monuments,  it  is 
remarkable  how  many  there  are  who  m.istake  altogether  the  duty  that 
now  devolves  upon  the  surveyor.  It  is  by  no  means  uncommon  that  we 
find  men  whose  theoretical  education  is  supposed  to  make  them  experts 
who  think  that  when  the  monuments  are  gone,  the  only  thing  to  be  done 
is  to  place  new  monuments  where  the  old  ones  should  have  been,  and 
where  they  would  have  been  if  placed  correctly.  This  is  a serious  mis- 
take. The  problem  is  now  the  same  that  it  was  before  : to  ascertain,  by 
the  best  lights  of  which  the  case  admits,  where  the  original  lines  were. 
The  mistake  above  alluded  to  is  supposed  to  have  found  expression  in 
our  legislation  ; though  it  is  possible  that  the  real  intent  of  the  act  to 
which  we  shall  refer  is  not  what  is  commonly  supposed. 

An  act  passed  in  1869,  Compiled  Laws,  § 593,  amending  the  laws  re- 
specting the  duties  and  powers  of  county  surveyors,  after  providing  for 
the  case  of  corners  which  can  be  identified  by  the  original  field-notes  or 
other  unquestionable  testimony,  directs  as  follows  ; 

“ Second.  Extinct  interior  section-corners  must  be  re-established  at 
the  intersection  of  two  right  lines  joining  the  nearest  known  points  on 
the  original  section  lines  east  and  west  and  north  and  south  of  it. 


APPENDIX  A. 


581 


''Third.  Any  extinct  quarter-section  corner,  except  on  fractional  lines, 
must  be  re-established  equidistant  and  in  a right  line  between  the  section 
corners;  in  all  other  cases  at  its  proportionate  distance  between  the 
nearest  original  corners  on  the  same  line.” 

The  corners  thus  determined,  the  surveyors  are  required  to  perpetu- 
ate by  noting  bearing  trees  when  timber  is  near. 

To  estimate  properly  this  legislation,  we  must  start  with  the  admit- 
ted and  unquestionable  fact  that  each  purchaser  from  government  bought 
such  land  as  was  within  the  original  boundaries,  and  unquestionably 
owned  it  up  to  the  time  when  the  monuments  became  extinct.  If  the 
monument  was  set  for  an  interior-section  corner,  but  did  not  happen  to 
be  “ at  the  intersection  of  two  right  lines  joining  the  nearest  known 
points  on  the  original  section  lines  east  and  west  and  north  and  south 
of  it.”  it  nevertheless  determined  the  extent  of  his  possessions,  and  he 
gained  or  lost  according  as  the  mistake  did  or  did  not  favor  him. 

It  will  probably  be  admitted  that  no  man  loses  title  to  his  land  or  any 
part  thereof  merely  because  the  evidences  become  lost  or  uncertain.  It 
may  become  more  difficult  for  him  to  establish  it  as  against  an  adverse 
claimant,  but  theoretically  the  right  remains;  and  it  remains  as  a poten- 
tial fact  so  long  as  he  can  present  better  evidence  than  any  other  person. 
And  it  may  often  happen  that,  notwithstanding  the  loss  of  all  trace  of  a 
section  corner  or  quarter  stake,  there  will  still  be  evidence  from  which  any 
surveyor  will  be  able  to  determine  with  almost  absolute  certainty  where 
the  original  boundary  was  between  the  government  subdivisions. 

There  are  two  senses  in  which  the  word  extinct  may  be  used  in  this 
connection  : one  the  sense  of  physical  disappearance  ; the  other  the 
sense  of  loss  of  all  reliable  evidence.  If  the  .statute  speaks  of  extinct 
corners  in  the  former  sense,  it  is  plain  that  a serious  mistake  was  made 
in  supposing  that  surveyors  could  be  clothed  with  authority  to  establish 
new  corners  by  an  arbitrary  rule  in  such  cases.  As  well  might  the  stat- 
ute declare  that  if  a man  lose  his  deed  he  shall  lose  his  land  altogether. 

But  if  by  extinct  corner  is  meant  one  in  respect  to  the  actual  location 
of  which  all  reliable  evidence  is  lost,  then  the  following  remarks  are  per- 
tinent; 

1.  There  would  undoubtedly  be  a presumption  in  such  a case  that 
the  corner  was  correctly  fixed  by  the  government  surveyor  where  the 
field -notes  indicated  it  to  be. 

2.  But  this  is  only  a presumption,  and  may  be  overcome  by  any  satis- 
factory evidence  showing  that  in  fact  it  was  placed  elsewhere. 

3.  No  statute  can  confer  upon  a county  surveyor  the  power  to  “estab- 
lish ” corners,  and  thereby  bind  the  parties  concerned.  Nor  is  this  a 
question  merely  of  conflict  between  State  and  Federal  law  ; it  is  a ques- 
tion of  property  right.  The  original  surv^eys  must  govern,  and  the  laws 
under  which  they  were  made  must  govern,  because  the  land  was  bought 
in  reference  to  them  ; and  any  legislation,  whether  State  or  Federal,  that 
should  have  the  effect  to  change  these,  would  be  inoperative,  because 
disturbing  vested  rights, 

4.  In  any  case  of  disputed  lines,  unless  the  parties  concerned  settle 
the  controversy  by  agreement,  the  determination  of  it  is  necessarily  a 


582 


S UR  VE  YING. 


judicial  act,  and  it  must  proceed  upon  evidence,  and  ^ivc  full  oppor- 
tunity for  a hcarinjT.  No  arl)itrary  rules  of  survey  or  of  evidence  can 
be  laid  down  whereby  it  can  be  adjudged. 

The  general  duty  of  a surveyor  in  such  a case  is  plain  enough.  lie 
is  not  to  assume  that  a monument  is  lost  until  after  he  has  thoroughly 
sifted  the  evidence  and  found  himself  unable  to  trace  it.  Kven  then  he 
should  hesitate  long  before  doing  anything  to  the  di.sturbance  of  settlerl 
possessions.  Occupation,  especially  if  long  continued,  often  alTords  very 
satisfactory  evidence  of  the  original  boundary  when  no  other  is  attain- 
able : and  the  surveyor  should  inquire  when  it  originated,  how,  and  why 
the  lines  were  then  located  as  they  were,  and  whether  a claim  of  title 
has  always  accompanied  the  po.ssession,  and  give  all  the  facts  due  force 
as  evidence.  Unfortunately,  it  is  known  that  surveyors  sometimes,  in 
supposed  obedience  to  the  State  statute,  disregard  all  evidences  of  occu- 
pation and  claim  of  title,  and  plunge  whole  neighborhoods  into  quarrels 
and  litigation  by  assuming  to  “establish  ” corners  at  points  with  which 
the  previous  occupation  cannot  harmonize.  It  is  often  the  case  that 
where  one  or  more  corners  are  found  to  be  extinct,  all  parties  concerned 
have  acquiesced  in  lines  which  were  traced  by  the  guidance  of  some 
other  corner  or  landmark,  which  may  or  may  not  have  been  trustworthy; 
but  to  bring  these  lines  into  discredit  when  the  people  concerned  do  not 
question  them  not  only  breeds  trouble  in  the  neighborhood,  but  it  must 
often  subject  the  surveyor  himself  to  annoyance  and  perhaps  discredit, 
since  in  a legal  controversy  the  law  as  well  as  common-sense  must  declare 
that  a supposed  boundary  line  long  acquiesced  in  is  better  evidence  of 
where  the  real  line  should  be  than  any  survey  made  after  the  original 
monuments  have  disappeared.  Stewart  vs.  Carleto?i,  Mich.  Reports, 
270;  Diehl  vs.  Zaiiger,  39  Mich.  Reports,  601  ; Dupont  vs.  Starrhig,  42 
Mich.  Reports,  492.  And  county  surveyors,  no  more  than  any  others, 
can  conclude  parties  by  their  surveys. 

The  mischiefs  of  overlooking  the  facts  of  possession  must  often  appear 
in  cities  and  villages.  In  towns  the  block  and  lot  stakes  soon  disappear; 
there  are  no  witness  trees  and  no  monuments  to  govern  except  such  as 
have  been  put  in  their  places,  or  where  their  places  were  supposed  to  be. 
The  streets  are  likely  to  be  soon  marked  off  by  fences,  and  the  lots  in  a 
block  will  be  measured  off  from  these,  without  looking  farther.  Now  it 
may  perhaps  be  known  in  a particular  case  that  a certain  monument  still 
remaining  was  the  starting-point  in  the  original  survey  of  the  town  plat; 
or  a surveyor  settling  in  tlie  town  may  take  some  central  point  as  the 
point  of  departure  in  his  surveys,  and  assuming  the  original  plat  to  be 
accurate,  he  will  then  undertake  to  find  all  streets  and  all  lots  by  course 
and  distance  according  to  the  plat,  measuring  and  estimating  from  his 
point  of  departure.  This  procedure  might  unsettle  every  line  and  every 
monument  existing  by  acquiescence  in  the  town  ; it  would  be  very  likely 
to  change  the  lines  of  streets,  and  raise  controversies  everywhere.  Yet 
this  is  what  is  sometimes  done  ; the  surveyor  himself  being  the  first 
person  to  raise  the  disturbing  questions. 

Suppose,  for  example,  a particular  village  street  has  been  located  by 
acquiescence  and  use  for  many  years,  and  the  proprietors  in  a certain 


APPENDIX  A. 


583 


block  have  laid  off  their  lots  in  reference  to  this  practical  location. 
Two  lot-owners  quarrel,  and  one  of  them  calls  in  a surveyor  that  he  may 
be  sure  that  his  neighbor  shall  not  get  an  inch  of  land  from  him.  This 
surveyor  undertakes  to  make  his  survey  accurate,  whether  the  original 
was,  or  not.  and  the  first  result  is,  he  notifies  the  lot-owners  that  there  is 
error  in  the  street  line,  and  that  all  fences  should  be  moved,  say,  one  foot 
to  the  east.  Perhaps  he  goes  on  to  drive  stakes  through  the  block  ac- 
cording to  this  conclusion.  Of  course,  if  he  is  right  in  doing  this,  all 
lines  in  the  village  will  be  unsettled  ; but  we  will  limit  our  attention  to 
the  single  block.  It  is  not  likely  that  the  lot-owners  generally  will  allow 
the  new  survey  to  unsettle  their  possessions,  but  there  is  always  a prob- 
ability of  finding  some  one  disposed  to  do  so.  We  shall  then  have  a 
lawsuit;  and  with  what  result.? 

It  is  a common  error  that  lines  do  not  become  fixed  by  acquiescence 
in  a less  time  than  twenty  years.  In  fact,  by  statute,  road  lines  maybe- 
come  conclusively  fixed  in  ten  years;  and  there  is  no  particular  time 
that  shall  be  required  to  conclude  private  owners,  where  it  appears  that 
they  have  accepted  a particular  line  as  their  boundary,  and  all  concerned 
have  cultivated  and  claimed  up  to  it.  McNamara  vs.  Seato?i,  82  111.  Re- 
ports, 498;  Biince  vs.  Bidivell,  43  Mich.  Reports,  542.  Public  policy  re- 
quires that  such  lines  be  not  lightly  disturbed,  or  disturbed  at  all  after 
the  lapse  of  any  considerable  time.  The  litigant,  therefore,  who  in  such 
a case  pins  his  faith  on  the  surveyor,  is  likely  to  suffer  for  his  reliance, 
and  the  surveyor  himself  to  be  mortified  by  a result  that  seems  to  im- 
peach his  judgment. 

Of  course  nothing  in  what  has  been  said  can  require  a surveyor  to 
conceal  his  own  judgment,  or  to  report  the  facts  one  way  w^hen  he  be- 
lieves them  to  be  another.  He  has  no  right  to  mislead,  and  he  may 
rightfully  express  his  opinion  that  an  original  monument  was  at  one 
place,  when  at  the  same  time  he  is  satisfied  that  acquiescence  has  fixed 
the  rights  of  parties  as  if  it  were  at  another.  But  he  would  do  mischief 
if  he  were  to  attempt  to  “ establish”  monuments  which  he  knew  would 
tend  to  disturb  settled  rights;  the  farthest  he  has  a right  to  go,  as  an 
officer  of  the  law,  is  to  express  his  opinion  where  the  monument  should 
be,  at  the  same  time  that  he  imparts  the  information  to  those  who  em- 
ploy him,  and  who  might  otherwise  be  misled,  that  the  same  authority 
that  makes  him  an  officer  and  entrusts  him  to  make  surveys,  also  allows 
parties  to  settle  their  own  boundary  lines,  and  considers  acquiescence  in 
a particular  line  or  monument,  for  any  considerable  period,  as  strong,  if 
not  conclusive,  evidence  of  such  settlement.  The  peace  of  the  com- 
munity absolutely  requires  this  rule.  Joyce  vs.  Williams,  26  Mich.  Re- 
ports, 332.  It  is  not  long  since  that,  in  one  of  the  leading  cities  of  the 
State,  an  attempt  was  made  to  move  houses  two  or  three  rods  into  a 
street,  on  the  ground  that  a survey  under  which  the  street  had  been 
located  for  many  years  had  been  found  on  more  recent  survey  to  be 
erroneous. 

From  the  foregoing  it  will  appear  that  the  duty  of  the  surveyor  where 
boundaries  are  in  dispute  must  be  varied  by  the  circumstances.  i.  He 
is  to  search  for  original  monuments,  or  for  the  places  where  they  were 


584 


SURVEYING, 


originally  located,  and  allow  these  to  control  if  he  finds  thetn,  unless  he 
has  reason  to  believe  tliat  agreements  of  the  parties,  express  or  implied, 
have  rendered  them  unimportant.  By  monuments  in  the  case  of  gov- 
ernment surveys  we  mean  of  course  the  corner  and  quarter  stakes: 
blazed  lines  or  marked  trees  on  the  lines  are  not  monuments;  tliey  are 
merely  guides  or  finger-posts,  if  we  may  use  tlie  expression,  to  inform  ns 
with  more  or  less  accuracy  where  the  monuments  may  be  found.  2.  If 
the  original  monuments  are  no  longer  discoverable,  the  question  of  loca- 
tion becomes  one  of  evidence  merely.  It  is  merely  idle  for  any  State 
statute  to  direct  a surveyor  to  locate  or  “establisli  ’ a corner,  as  the  place 
of  the  original  monument,  according  to  some  inflexible  rule.  The  sur- 
veyor on  the  other  hand  must  inquire  into  all  the  facts  ; giving  due  prom- 
inence to  the  acts  of  parties  concerned,  and  always  keeping  in  mind, 
first,  that  neither  his  opinion  nor  his  survey  can  be  conclusive  upon 
parties  concerned  ; second,  tliat  courts  and  juries  may  be  required  to  fol- 
low after  the  surveyor  over  the  same  ground,  and  that  it  is  exceedingly 
desirable  that  he  govern  his  action  by  the  same  lights  and  rules  that  will 
govern  theirs.  On  town  plats  if  a surplus  or  deficiency  appears  in  a 
block,  when  the  actual  boundaries  are  compared  with  the  original  figures, 
and  there  is  no  evidence  to  fix  the  exact  location  of  the  stakes  which 
marked  the  division  into  lots,  the  rule  of  common-sense  and  of  law  is 
that  the  surplus  or  deficiency  is  to  be  apportioned  between  the  lots,  on 
an  assumption  that  the  error  extended  alike  to  all  parts  of  the  block. 
O' Brien  vs.  McGra7ie,  29  Wis.  Reports,  446 ; Qumnm  vs.  Reixers,  46 
Mich.  Reports,  605. 

It  is  always  possible  when  corners  are  extinct  that  the  surveyor  may 
usefully  act  as  a mediator  between  parties,  and  assist  in  preventing  legal 
controversies  by  settling  doubtful  lines.  Unless  he  is  made  for  this  pur- 
pose an  arbitrator  by  legal  submission,  the  parties,  of  course,  even  if  they 
consent  to  follow  his  judgment,  cannot,  on  the  basis  of  mere  consent,  be 
compelled  to  do  so:  but  if  he  brings  about  an  agreement,  and  they  carry 
it  into  effect  by  actually  conforming  their  occupation  to  his  lines,  the 
action  will  conclude  them.  Of  course  it  is  desirable  that  all  such  agree- 
ments be  reduced  to  writing;  but  this  is  not  absolutely  indispensable  if 
they  are  carried  into  effect  without. 

Meander  Lines.  — The  subject  to  which  allusion  will  now  be  made  is 
taken  up  with  some  reluctance,  because  it  is  believed  the  general  rules 
are  familiar.  Nevertheless  it  is  often  found  that  surveyors  misapprehend 
them,  or  err  in  their  application;  and  as  other  interesting  topics  are 
somewhat  connected  with  this,  a little  time  devoted  to  it  will  probably 
not  be  altogether  lost.  The  subject  is  that  of  meander  lines.  These 
are  lines  traced  along  the  shores  of  lakes,  ponds,  and  considerable  rivers 
as  the  measures  of  quantity  when  sections  are  made  fractional  by  such 
waters.  These  have  determined  the  price  to  be  paid  when  government 
lands  were  bought,  and  perhaps  the  impression  still  lingers  in  some 
minds  that  the  meander  lines  are  boundary  lines,  and  all  in  front  of 
them  remains  unsold.  Of  course  this  is  erroneous.  There  was  never 
any  doubt  that,  except  on  the  large  navigable  rivers,  the  boundary  of  the 
owners  of  the  banks  is  the  middle  line  of  the  river;  and  while  some 


APPENDIX  A. 


585 


courts  have  held  that  this  was  the  rule  on  all  fresh-water  streams,  large 
and  small,  others  have  held  to  the  doctrine  that  the  title  to  the  bed  of  the 
stream  below  low-water  mark  is  in  the  State,  while  conceding  to  the 
owners  of  the  banks  all  riparian  rights.  The  practical  difference  is  not 
very  important.  In  this  State  the  rule  that  the  centre  line  is  the  bound- 
ary line  is  applied  to  all  our  great  rivers,  including  the  Detroit,  varied 
somewhat  by  the  circumstance  of  there  being  a distinct  channel  for 
navigation  in  some  cases  with  the  stream  in  the  main  shallow,  and  also 
sometimes  by  the  existence  of  islands. 

The  troublesome  questions  for  surveyors  present  themselves  when  the 
boundary  line  between  two  contiguous  estates  is  to  be  continued  from  the 
meander  line  to  the  centre  line  of  the  river.  Of  course  the  original  sur- 
vey supposes  that  each  purchaser  of  land  on  the  stream  has  a water-front 
of  the  length  shown  by  the  field-notes ; and  it  is  presumable  that  he 
bought  this  particular  land  because  of  that  fact.  In  many  cases  it  now 
happens  that  the  meander  line  is  left  some  distance  from  the  shore  by 
the  gradual  change  of  course  of  the  stream  or  diminution  of  the  flow 
of  water.  Now  the  dividing  line  between  two  government  subdivisions 
might  strike  the  meander  line  at  right  angles,  or  obliquely  ; and  in  some 
cases,  if  it  were  continued  in  the  same  direction  to  the  centre  line  of  the 
river,  might  cut  off  from  the  water  one  of  the  subdivisions  entirely,  or  at 
least  cut  it  off  from  any  privilege  of  navigation,  or  other  valuable  use 
of  the  water,  while  the  other  might  have  a water-front  much  greater 
than  the  length  of  a line  crossing  it  at  right  angles  to  its  side  lines. 
The  effect  might  be  that,  of  two  government  subdivisions  of  equal  size 
and  cost,  one  would  be  of  very  great  value  as  water-front  property,  and 
the  other  comparatively  valueless.  A rule  which  would  produce  this  re- 
sult would  not  be  just,  and  it  has  not  been  recognized  in  the  law. 

Nevertheless  it  is  not  easy  to  determine  what  ought  to  be  the  correct 
rule  for  every  case.  If  the  river  has  a straight  course,  or  one  nearly  so, 
every  man’s  equities  will  bp  preserved  by  this  rule  ; Extend  the  line  of 
division  between  the  two  parcels  from  the  meander  line  to  the  centre  line 
of  the  river,  as  nearly  as  possible  at  right  angles  to  the  general  course  of 
the  river  at  that  point.  This  will  preserve  to  each  man  the  water  front 
which  the  field-notes  Indicated,  except  as  changes  in  the  water  may  have 
affected  it,  and  the  only  inconvenience  will  be  that  the  division  line  be- 
tween different  subdivisions  is  likely  to  be  more  or  less  deflected  where  it 
strikes  the  meander  line. 

This  is  the  legal  rule,  and  it  is  not  limited  to  government  surveys,  but 
applies  as  well  to  water  lots  which  appear  as  such  on  town  plats.  Bay 
City  Gas  Light  Co.  v.  The  I Jidnstrial  Works,  28  Mich.  Reports,  182.  It 
often  happens,  therefore,  that  the  lines  of  city  lots  bounded  on  navigable 
streams  are  deflected  as  they  strike  the  bank,  or  the  line  where  the  bank 
was  when  the  town  was  first  laid  out. 

When  the  stream  is  very  crooked,  and  especially  if  there  are  short 
bends,  so  that  the  foregoing  rule  is  incapable  of  strict  application,  it  is 
sometimes  very  difficult  to  determine  what  shall  be  done;  and  in  many 
cases  the  surveyor  may  be  under  the  necessity  of  working  out  a rule  for 
himself.  Of  course  his  action  cannot  be  conclusive;  but  if  he  adopts  one 


586 


SU/^  VE  YING. 


that  follows,  as  nearly  as  the  circumstances  will  admit,  the  j:^cncral  rule 
above  indicated,  so  as  to  divide  as  near  as  may  be  the  bed  of  the  stream 
amonj^  the  atljoininj^  owners  in  proportion  to  their  lines  u[)on  the  shore, 
his  division,  beinp^  that  of  an  expert,  macle  upon  the  ground  aiul  with  all 
available  lights,  is  likely  to  be  adopted  as  law  for  the  case.  Judicial  de- 
cisions, into  which  the  surveyor  would  find  it  ])rudent  to  look  under  such 
circuitistances,  will  throw  lit(ht  upon  his  duties  anrl  may  constitute  a suf- 
ficient ^uide  when  peculiar  cases  arise.  Each  riparian  lot-owner  ought  to 
have  a line  on  the  legal  boundary,  namely,  the  centre  line  of  the  stream, 
proportioned  to  the  length  of  his  line  on  the  shore;  and  the  problem  in 
each  case  is,  how  this  is  to  be  given  him.  Alluvion,  when  a river  imper- 
ceptibly changes  its  course,  will  be  apportioned  by  the  same  rules. 

The  existence  of  islands  in  a stream,  when  the  middle  line  constitutes 
a boundary,  will  not  affect  the  apportionment  unless  the  islands  were 
surveyed  out  as  government  subdivisions  in  the  original  admeasurement. 
Wherever  that  was  the  case,  the  purchaser  of  the  island  divides  the  bed 
of  the  stream  on  each  side  with  the  owner  of  the  bank,  and  his  rights 
also  extend  above  and  below  the  solid  ground,  and  are  limited  by  the 
peculiarities  of  the  bed  and  the  channel.  If  an  islanrl  was  not  surveyed  as 
a government  subdivision  previous  to  the  sale  of  the  bank,  it  is  of  course 
impossible  to  do  this  for  the  purposes  of  government  sale  afterwards,  for 
the  reason  that  the  rights  of  the  bank  owners  are  fixed  by  their  purchase: 
when  making  that,  they  have  a right  to  understand  that  all  land  between 
the  meander  lines,  not  separately  surveyed  and  sold,  will  pass  with  the 
shore  in  the  government  sale  ; and  having  this  right,  anything  which 
their  purchase  would  include  under  it  cannot  afterward  be  taken  from 
them.  It  is  believed,  however,  that  the  federal  courts  would  not  recog- 
nize the  applicability  of  this  rule  to  large  navigable  rivers,  such  as  those 
uniting  the  great  lakes. 

On  all  the  little  lakes  of  the  State  which  are  mere  expansions  near 
their  mouths  of  the  rivers  passing  through  them — such  as  the  Muskegon, 
Pere  Marquette  and  Manistee — the  same  rule  of  bed  ownership  has  been 
judicially  applied  that  is  applied  to  the  rivers  themselves ; and  the  divi- 
sion lines  are  extended  under  the  water  in  the  same  way.  Rice  v.  Ruddi- 
man,  lo  Mich.,  125.  If  such  a lake  were  circular,  the  lines  would  con- 
verge to  the  centre:  if  oblong  or  irregular,  there  might  be  a line  in  the 
middle  on  which  they  would  terminate,  whose  course  would  bear  some 
relation  to  that  of  the  shore.  But  it  can  seldom  be  important  to  follow 
the  division  line  very  far  under  the  water,  since  all  private  rights  are  sub- 
ject to  the  public  rights  of  navigation  and  other  use,  and  any  private  use 
of  the  lands  inconsistent  with  these  would  be  a nuisance,  and  punishable 
as  such.  It  is  sometimes  important,  however,  to  run  the  lines  out  for 
some  considerable  distance,  in  order  to  determine  where  one  may  law- 
fully moor  vessels  or  rafts,  for  the  winter,  or  cut  ice.  The  ice  crop  that 
forms  over  a man’s  land  of  course  belongs  to  him.  Lo7'ina7i  v.  Be7iso7i,  8 
Mich.,  18  ; People  s Ice  Co.  v.  Stea777er  Excelsior,  recently  decided. 

What  is  said  above  will  show  how  unfounded  is  the  notion,  which  is 
sometimes  advanced,  that  a riparian  proprietor  on  a meandered  river  may 
lawfully  raise  the  water  in  the  stream  without  liability  to  the  proprietors 


APPENDIX  A. 


587 


above,  provided  he  does  not  raise  it  so  that  it  overflows  the  meander  line. 
The  real  fact  is  that  the  meander  line  has  nothing  to  do  with  such  a case, 
and  an  action  will  lie  whenever  he  sets  back  the  water  upon  the  proprie- 
tor above,  whether  the  overflow  be  below  the  meander  lines  or  above 
them. 

As  regards  the  lakes  and  ponds  of  the  State,  one  may  easily  raise 
questions  that  it  would  be  impossible  for  him  to  settle.  Let  us  suggest 
a few  questions,  some  of  which  are  easily  answered,  and  some  not : 

1.  To  whom  belongs  the  land  under  these  bodies  of  water,  where  they 
are  not  mere  expansions  of  a stream  flowing  through  them  ? 

2.  What  public  rights  exist  in  them  ? 

3.  If  there  are  islands  in  them  which  were  not  surveyed  out  and  sold 
by  the  United  States,  can  this  be  done  now.? 

Others  will  be  suggested  by  the  answers  given  to  these. 

It  seems  obvious  that  the  rules  of  private  ownership  which  are  applied 
to  rivers  cannot  be  applied  to  the  great  lakes.  Perhaps  it  should  be  held 
that  the  boundary  is  at  low-water  mark,  but  improvements  beyond  this 
would  only  become  unlawful  when  they  became  nuisances.  Islands  in 
the  great  lakes  would  belong  to  the  United  States  until  sold,  and  might 
be  surveyed  and  measured  for  sale  at  any  time.  The  right  to  take  fish  in 
the  lakes,  or  to  cut  ice,  is  public  like  the  right  of  navigation,  but  is  to  be 
exercised  in  such  manner  as  not  to  interfere  with  the  rights  of  shore 
owners.  But  so  far  as  these  public  rights  can  be  the  subject  of  ownership, 
they  belong  to  the  State,  not  to  the  United  States ; and,  so  it  is  believed, 
does  the  bed  of  a lake  also.  Pollard  v.  Hagan,  3 Howard’s  U.  S.  Reports. 
But  such  rights  are  not  generally  considered  proper  subjects  of  sale,  but, 
like  the  right  to  make  use  of  the  public  highways,  they  are  held  by  the 
State  in  trust  for  all  the  people. 

What  is  said  of  the  large  lakes  may  perhaps  be  said  also  of  many  of 
the  interior  lakes  of  the  State ; such,  for  example,  as  Houghton,  Higgin.s, 
Cheboygan,  Burt’s,  Mullet,  Whitmore,  and  many  others.  But  there  are 
many  little  lakes  or  ponds  which  are  gradually  disappearing,  and  the 
shore  pr opi  ietor ship  advances  pari  j^>assu  as  the  waters  recede.  If  these 
are  of  any  considerable  size — say,  even  a mile  across — there  may  be  ques- 
tions of  conflicting  rights  which  no  adjudication  hitherto  made  could 
settle.  Let  any  surveyor,  for  example,  take  the  case  of  a pond  of  irregu- 
lar form,  occupying  a mile  square  or  more  of  territory,  and  undertake  to 
determine  the  rights  of  the  shore  proprietors  to  its  bed  when  it  shall 
totally  disappear,  and  he  will  find  he  is  in  the  midst  of  problems  such  as 
probably  he  has  never  grappled  with,  or  reflected  upon  before.  But  the 
general  rules  for  the  extension  of  shore  lines,  which  have  already  been 
laid  down,  should  govern  such  cases,  or  at  least  should  serve  as  guides  in 
their  settlmeent.  Hole. — Since  this  address  was  delivered  some  of  these 
questions  have  received  the  attention  of  the  Supreme  Court  of  Michigan 
m the  cases  of  Richardson.v,  Prentiss,  48  Mich.  Reports,  88,  and  Backus 
V.  Detroit,  Albany  Law  Journal,  vol,  26,  p.  428, 

Where  a pond  is  so  small  as  to  be  included  within  the  lines  of  a pri- 
vate purchase  from  the  government,  it  is  not  believed  the  public  have  any 
rights  in  it  whatever.  Where  it  is  not  so  included,  it  is  believed  they  have 


588 


SUR  VE  YING. 


rights  of  fishery,  rights  to  take  ice  and  water,  and  rights  of  navigation  for 
business  or  pleasure.  Tliis  is  tlie  common  belief,  atifl  probably  the  just 
one.  Shore  rights  must  not  be  so  exercised  as  to  disturb  these,  and  tlie 
States  may  pass  all  proper  laws  for  their  protection.  It  would  be  easy 
with  suitable  legislation  to  preserve  these  litlle  bodies  of  water  as  perma- 
nent places  of  resort  for  the  pleasure  and  recreation  of  the  people,  and 
there  ought  to  be  such  legislation. 

If  the  State  should  be  recognized  as  owner  of  the  beds  of  these  small 
lakes  and  ponds,  it  would  not  be  owner  for  the  purpose  of  selling.  It 
would  be  owner  only  as  a trustee  for  the  public  use;  and  a sale  would  be 
inconsistent  with  the  right  of  the  bank  owners  to  make  use  of  the  water 
in  its  natural  condition  in  connection  with  their  estates.  Some  of  them 
might  be  made  salable  lands  by  draining  ; but  the  State  could  not  drain, 
even  for  this  purpose,  against  the  will  of  the  shore  owners,  unless  their 
rights  were  appropriated  and  paid  for. 

Upon  many  questions  that  might  arise  between  the  State  as  owner  of 
the  bed  of  a little  lake  and  the  shore  owners,  it  would  be  presumptuous  to 
express  an  opinion  now,  and  fortunately  the  occasion  does  not  require  it. 

I have  thus  indicated  a few  of  the  questions  with  which  surveyors  may 
nowand  then  have  occasion  to  deal,  and  to  which  they  should  bring  good 
sense  and  sound  judgment.  Surveyors  are  not  and  cannot  be  judicial 
officers,  but  in  a great  many  cases  they  act  in  a gtiasi  judicial  capacity 
with  the  acquiescence  of  parties  concernerJ  : and  it  is  important  for  them 
to  know  by  what  rules  they  are  to  be  guided  in  the  discharge  of  their 
judicial  functions.  What  I have  said  cannot  contribute  much  to  their 
enlightenment,  but  I trust  will  not  be  wholly  without  value. 


APPENDIX  B. 


INSTRUCTIONS  TO  U.  S.  DEPUTY  MINERAL  SURVEYORS. 
FOR  THE  DISTRICT  OF  COLORADO.  (1886.) 

GENERAL  RULES. 

1.  All  official  communications  must  be  addressed  to  the  Surveyor-Gen- 
eral. You  will  always  refer  to  the  date  and  subject-matter  of  the  letter 
to  which  you  reply,  and  when  a mineral  claim  is  the  subject  of  corre- 
spondence, you  will  give  the  name,  ownership  and  survey  number. 

2.  You  should  keep  a complete  record  of  each  survey  made  by  you, 
and  the  facts  coming  to  your  knowledge  at  the  time,  as  well  as  copies  of 
all  your  field-notes,  reports  and  official  correspondence,  in  order  that 
such  evidence  may  be  readily  produced  when  called  for  at  any  future 
time. 

3.  Field-notes  and  other  reports  must  be  written  in  a clear  and  legible 
hand,  and  upon  the  proper  blanks  furnished  by  this  office.  No  cut  sheets, 
interlineations  or  erasures  will  be  allowed  ; and  no  abbreviations  or  sym- 
bols must  be  used,  except  such  as  are  indicated  in  the  specimen  field- 
notes. 

4.  No  return  by  you  will  be  recognized  as  official  unless  made  in  pur- 
suance of  a special  order  from  this  office. 

5.  The  claimant  is  required,  in  all  cases,  to  make  satisfactory  arrange- 
ments with  you  for  the  payment  for  your  services  and  those  of  your 
assistants  in  making  the  survey,  as  the  United  States  will  not  be  held 
responsible  for  the  payment  of  the  same.  You  will  call  the  attention  of 
applicants  for  mineral-survey  orders  to  the  requirements  of  the  circular 
of  this  date  in  the  appendix. 

6.  You  will  promptly  notify  this  office  of  any  change  in  your  post-office 
address.  Upon  permanent  removal  from  the  State,  you  are  expected  to 
resign  your  appointment. 

NOT  TO  ACT  AS  ATTORNEY. 

7.  You  are  precluded  from  acting,  either  directly  or  indirectly,  as  at- 
torney in  mineral  claims.  Your  duty  in  any  particular  case  ceases  when 
you  have  executed  the  survey  and  returned  the  field-notes  and  prelimi- 
nary plat,  with  your  report  to  the  Surveyor-General.  You  will  not  be  al- 
lowed to  prepare  for  the  mining  claimant  the  papers  in  support  ol  his 


590 


VE  YING. 


application  for  patent,  or  otherwise  perform  the  duties  of  nn  attorney 
before  the  land  office  in  connection  with  a mininjr  claim.  You  arc  not 
permitted  to  combine  the  duties  of  surveyor  and  notary-public  in  tlic 
same  case  by  administering  oaths  to  the  parties  in  interest.  In  short, 
you  must  have  absolutely  nothing  to  do  with  the  case  except  in  your 
official  capacity  as  surveyor.  You  will  make  no  survey  of  a mineral 
claim  in  which  you  hold  an  interest. 

THE  FIELD-WORK. 

8.  The  survey  made  and  reported  must,  in  every  case,  be  an  actual  sur-  , 
vey  on  the  ground  in  full  detail,  made  by  you  in  person  after  the  receipt 
of  the  order,  and  without  reference  to  any  knowledge  you  may  have  pre- 
viously acquired  by  reason  of  having  made  the  location-survey  or  other- 
wise, and  must  show  the  actual  facts  existing  at  the  time.  If  the  season 
of  the  year,  or  any  other  cause,  renders  such  personal  examination  im- 
possible, you  will  postpone  the  survey,  and  under  no  circumstances  rely 
upon  the  statements  or  surveys  of  other  parties,  or  upon  a former  exami- 
nation by  yourself. 

The  term  survey  in  these  instructions  applies  not  only  to  the  usual 
field-work,  but  also  to  the  examinations  required  for  the  preparation  of 
your  affidavits  of  five  hundred  dollars  expenditure,  descriptive  reports  on 
placer-claims  and  all  other  reports. 

SURVEY  AND  LOCATION. 

9.  The  survey  must  be  made  in  strict  conformity  with,  or  be  embraced 
within,  the  lines  of  the  record  of  location  upon  which  the  order  is  based. 

If  the  survey  and  location  are  identical,  that  fact  must  be  clearlv  and 
distinctly  stated  in  your  field-notes.  If  not  identical,  a bearing  and  dis- 
tance must  be  given  from  each  established  corner  of  the  survey  to  the 
corresponding  corner  of  the  location.  The  lines  of  the  location,  as 
found  upon  the  ground,  must  be  laid  down  upon  the  preliminary  plat  in 
such  manner  as  to  contrast  and  show  their  relation  to  the  lines  of  the 
survey. 

10.  If  the  record  of  location  has  been  made  prior  to  the  passage  of  the 
mining  act  of  May  10,  1872,  and  is  not  sufficiently  definite  and  certain  to 
enable  you  to  make  a correct  survey  therefrom,  you  are  required,  after 
reasonable  notice  in  writing,  to  be  served  personally  or  through  the 
United  States  mail  on  the  applicant  for  survey  and  adjoining  claimants, 
whose  residence  or  post-office  address  you  may  know,  or  can  ascertain  by 
the  exercise  of  reasonable  diligence,  to  take  testimony  of  neighboring 
claimants  and  other  persons  who  are  familiar  with  the  boundaries  there- 
of as  originally  located  and  asserted  by  the  locators  of  the  claim,  and 
after  having  ascertained  by  such  testimony  the  boundaries  as  originally 
established,  you  will  make  a survey  in  accordance  therewith,  and  trans- 
mit full  and  correct  returns  of  the  survey,  accompanied  by  the  copy  of 
the  record  of  location,  the  testimony,  and  a copy  of  the  notice  served  on 
the  claimant  and  adjoining  proprietors,  certifying  thereon  when,  in  what 
manner,  and  on  whom  service  was  made. 

11.  If  the  location  has  been  made  subsequent  to  the  passage  of  the 


APPENDIX  B. 


591 


mining  act  of  May  10,  1872,  and  the  law  has  been  complied  with  in  the  mat- 
ter of  marking  the  location  on  the  ground  and  recording  the  same,  and 
any  question  should  arise  in  the  execution  of  the  survey  as  to  the  iden- 
tity of  monuments,  marks,  or  boundaries  which  cannot  be  determined  by 
a reference  to  the  record,  you  are  required  to  take  testimony  in  the  man- 
ner hereinbefore  prescribed  for  surveys  of  claims  located  prior  to  May  10, 
1872,  and  having  thus  ascertained  the  true  and  correct  boundaries  origi- 
nally established,  marked  and  recorded,  you  will  make  the  survey  accord- 
ingly. 

12.  In  accordance  with  the  principle  that  courses  and  distances  must 
give  way  when  in  conflict  with  fixed  objects  and  monuments,  you  will 
not,  under  any  circumstances,  change  the  corners  of  the  location  for  the 
purpose  of  making  them  conform  to  the  description  in  the  record.  If 
the  difference  from  the  location  be  slight,  it  may  be  explained  in  the  field- 
notes,  but  if  there  should  be  a wide  discrepancy,  you  will  report  the  facts 
to  this  office  and  await  further  instructions. 

INSTRUMENT. 

13.  All  mineral  surveys  must  be  made  with  a SOLAR  transit,  or 
other  instrument  operating  independently  of  the  magnetic  needle,  and 
all  courses  must  be  referred  to  the  true  meridian.  It  is  deemed  best 
that  a solar  transit  should  be  used  under  all  circumstances.  The  varia- 
tion should  be  noted  at  each  corner  of  the  survey. 

CONNECTIONS. 

14.  Connect  corner  No.  i of  your  survey  by  course  and  distance  with 
some  corner  of  the  public  survey  or  with  a United  States  location-mon- 
ument, if  the  claim  lies  within  two  miles  of  such  corner  or  monument. 
If  both  are  within  the  required  distance,  you  will  connect  with  the  near- 
est corner  of  the  public  survey. 

LOCATION-MONUMENTS. 

15.  In  case  your  survey  is  situated  in  a district  where  there  are  no 
corners  of  the  public  survey  and  no  monuments  within  the  prescribed 
limits,  you  will  proceed  to  establish  a mineral  monument,  in  the  location 
of  which  you  will  exercise  the  greatest  care  to  insure  permanency  as  to 
site  and  construction. 

16.  The  site,  when  practicable,  should  be  some  prominent  point  visi- 
ble for  a long  distance  from  every  direction,  and  should  be  so  chosen 
that  the  permanency  of  the  monument  will  not  be  endangered  by  snow, 
rock  or  land  slides,  or  other  natural  causes. 

17.  The  location-monument  should  consist  of  a post  eight  feet  long 
and  six  inches  square  set  three  feet  in  the  ground,  and  protected  by  a 
well-built  conical  mound  of  stone  three  feet  high  and  six  feet  base. 
The  letters  U.  S.  L.  M.  followed  by  a na7ne  must  be  scribed  on  the  post 
and  also  chiselled  on  a large  stone  in  the  mound,  or  on  the  rock  in  place 
that  may  form  the  base  of  the  monument.  There  is  no  objection  to  the 
establishment  of  a location-monument  of  larger  size,  or  of  other  material 
of  equally  durable  character. 

18.  From  the  monument,  connections  by  course  and  distance  must 


592 


SUJ^  VE  YING. 


be  taken  to  two  or  three  bearing  trees  or  rocks,  and  to  any  well-known 
natural  and  permanent  objects  in  the  vicinity,  such  as  the  confluence  of 
streams,  prominent  rocks,  buildings,  shafts  or  mouths  of  adits.  Bearings 
should  also  be  taken  to  prominent  mountain-peaks,  and  the  approximate 
distance  and  direction  ascertained  from  the  nearest  town  or  mining 
camp.  A detailed  description  of  the  location-monument  must  be  in- 
cluded in  the  field- notes  of  the  survey  for  which  it  is  established. 

CORNERS. 

19.  Corners  may  consist  of 

First — A stone  at  least  twenty-four  inches  long  by  six  inches  square 
set  eighteen  inches  in  the  ground. 

Second— K post  at  least  four  and  a half  feet  long  by  four  inches 
square  set  twelve  inches  in  the  ground  and  surrounded  by  a mound  of 
stone  or  earth  two  and  a half  feet  high  and  five  feet  base. 

Third — A rock  in  place. 

20.  All  corners  must  be  established  in  a permanent  and  workmanlike 
manner,  and  the  corner  and  survey  number  must  be  neatly  chiselled  or 
scribed  on  the  sides  facing  the  claim.  When  a rock  in  place  is  used  its 
dimensions  above  ground  must  be  stated,  and  a cross  chiselled  at  the  ex- 
act corner-point. 

21.  In  case  the  point  for  the  corner  be  inaccessible  or  unsuitable,  you 
will  establish  a witness-corner,  which  must  be  marked  with  the  letters 
W.  C.  in  addition  to  the  corner  and  survey  number.  The  witness-corner 
should  be  located  upon  a line  of  the  survey  and  as  near  as  practicable  to 
the  true  corner,  with  which  it  must  be  connected  by  course  and  distance. 
The  reason  for  the  establish m.ent  of  a witness-corner  must  always  be 
stated  in  the  field-notes. 

22.  The  identity  of  all  corners  should  be  perpetuated  by  taking 
courses  and  distances  to  bearing  trees,  rocks,  and  other  objects,  as  pre- 
scribed in  the  establishment  of  location-monuments.  If  an  official  sur- 
vey has  been  made  within  a reasonable  distance  in  the  vicinity,  you  will 
run  a connecting  line  to  some  corner  of  the  same,  and  connect  in  like 
manner  with  all  conflicting  surveys  and  claims. 

TOPOGRAPHY. 

23.  Note  carefully  all  topographical  features  of  the  claim,  taking  dis- 
tances on  your  lines  to  intersections  with  all  streams,  gulches,  ditches, 
ravines,  mountain  ridges,  roads,  trails,  etc.,  with  their  widths,  courses  and 
other  data  that  may  be  required  to  map  them  correctly.  If  the  claim 
lies  within  a town-site,  locate  all  municipal  improvements,  such  as  blocks, 
streets  and  buildings. 

24.  You  are  required  also  to  locate  all  mining  and  other  improve- 
ments upon  the  claim  by  courses  and  distances  from  corners  of  the  sur- 
vey, or  by  rectangular  offsets  from  tlie  centre  line,  specifying  the  dimen- 
sions and  character  of  each  in  full  detail. 

CONFLICT.S. 

25.  If  in  running  the  exterior  boundaries  of  a claim,  you  find  that  two 
surveys  conflict,  you  will  determine  the  courses  and  distances  from  the 


APPENDIX  B. 


593 


established  corners  at  which  the  exterior  boundaries  of  the  respective 
surveys  intersect  each  other,  and  run  all  lines  necessary  for  the  determi- 
nation of  the  areas  in  conflict,  both  with  surveyed  and  unsurveyed 
claims.  You  are  not  required,  however,  to  show  conflicts  with  unsur- 
veyed claims  unless  the  same  are  to  be  excluded. 

26.  When  a placer-claim  includes  lodes,  or  when  several  lode-loca- 
tions are  included  as  one  claim  in  one  survey,  you  will  preserve  a con- 
secutive series  of  numbers  for  the  corners  of  the  whole  survey  in  each 
case.  In  the  former  case  you  will  first  describe  the  placer-claim  in  your 
field-notes. 

PLACER-CLAIMS  AND  MILL-SITES. 

27.  The  exterior  lines  of  placer-claims  cannot  be  extended  over  other 
claims,  and  the  conflicting  areas  excluded  as  with  lode-claims,  it  being 
the  surface  ground  only,  with  side  lines  taken  perpendicularly  downward 
for  which  application  is  made.  The  survey  must  accurately  define  the 
boundaries  of  the  claim.  The  same  rule  will  apply  to  the  survey  of  mill- 
sites. 

28.  If  by  reason  of  intervening  surveys  or  claims  a placer  or  mill-site 
survey  should  be  divided  into  separate  tracts,  you  will  also  preserve  a 
consecutive  series  of  numbers  for  the  corners  of  the  whole  survey,  and 
distinguish  the  detached  portions  as  Lot  No.  i.  Lot  No.  2,  etc.,  connect- 
ing by  course  and  distance  a corner  of  each  lot  with  some  corner  of  the 
one  previously  described. 

LODE  AND  MILL-SITE. 

29.  A lode  and  mill-site  claim  in  one  survey  will  be  distinguished  by 
the  letters  A and  B following  the  number  of  the  survey.  The  corners  of 
the  mill-site  will  be  numbered  independently  of  those  of  the  lode.  Cor- 
ner No.  I of  the  mill-site  must  be  connected  with  a corner  of  the  lode 
claim  as  well  as  with  a corner  of  the  public  survey  or  U.  S.  location- 
monument. 

FIELD-NOTES. 

30.  In  order  that  the  results  of  your  survey  may  be  reported  to  this 
office  in  a uniform  manner,  you  will  prepare  your  field-notes  and  pre- 
liminary plat  in  strict  conformity  with  the  specimen  field-notes  and  plat, 
which  are  made  part  of  these  instructions.  They  are  designed  to  furnish 
you  with  all  needed  information  concerning  the  manner  of  describing 
the  boundaries,  corners,  connections,  intersections,  conflicts  and  improve- 
ments, and  stating  the  variation,  area,  location  and  other  data  con- 
nected with  the  survey  of  mineral  claims,  and  contain  forms  of  affidavits 
for  the  deputy-surveyor  and  his  assistants. 

In  your  first  reference  to  any  other  mineral  claim  you  will  give  the 
name,  ownership,  and  if  surveyed,  the  survey-number. 

31.  The  total  area  of  a lode-claim  embraced  by  the  exterior  bounda- 
ries, and  also  the  area  in  conflict  with  each  intersecting  survey  or  claim 
should  be  so  stated,  that  the  conflicts  with  any  one  or  all  of  them  may 
be  included  or  excluded  from  your  survey.  'I'his  will  enable  the  claim- 
ant to  state  in  his  application  for  patent  the  portions  to  be  excluded  in 
express  terms,  and  to  readily  determine  the  net  area  of  his  claim. 


594 


SURVEYING. 


32.  You  will  state  particularly  whether  the  claim  is  upon  surveyed  or 
unsurveyed  public  lands.  J?iving  in  the  former  case  the  quarter-section, 
township  and  range  in  which  it  is  located,  and  in  the  latter  the  township, 
as  near  as  can  be  determined. 

33.  The  field-notes  must  contain  the  post-office  address  of  the  claim- 
ant or  his  authorized  agent. 

EXPENDITURE  OF  FIVE  HUNDRED  DODLAR.S. 

34.  The  claimant  is  required  by  law,  either  at  the  tim^  of  filing  his 
application,  or  at  any  time  thereafter,  within  the  sixty  days  of  publica- 
tion, to  file  with  the  Register  the  certificate  of  the  Surveyor-General  that 
five  hundred  dollars’  worth  of  labor  has  been  expended  or  improvements 
made  upon  the  claim  by  himself  or  grantors.  The  information  upon 
which  to  base  this  certificate  must  be  derived  from  the  deputy  who 
makes  the  actual  survey  and  examination  upon  the  premises,  and  such 
deputy  is  required  to  specify  with  particularity  and  full  detail  the  char- 
acter and  extent  of  such  improvements.  See  also  Sec.  8. 

35.  When  a survey  embraces  several  locations  or  claims  held  in  com- 
mon, constituting  one  entire  claim,  whether  lode  or  placer,  an  expendi- 
ture of  five  hundred  dollars  upon  such  entire  claim  embraced  in  the  sur- 
vey will  be  sufficient  and  need  not  be  shown  upon  each  of  the  locations 
included  therein. 

36.  In  case  of  a lode  and  mill-site  claim  in  the  same  survey,  an  ex- 
penditure of  five  hundred  dollars  must  be  shown  upon  the  lode-claim 
only. 

37.  Only  actual  expenditures  and  mming  improvements,  made  by  the 
claimant  or  his  grantors,  having  a direct  relation  to  the  development  of 
the  claim,  can  be  included  in  your  estimate. 

38.  The  expenditures  required  may  be  made  from  the  surface,  or  in 
running  a tunnel  for  the  development  of  the  claim.  Improvements  of 
any  other  character,  such  as  buildings,  machinery  or  roadway.s,  must  be 
excluded  from  your  estimate  unless  you  show  clearly  that  they  are  asso- 
ciated with  actual  excavations,  such  as  cuts,  tunnels,  shafts,  etc.,  and  are 
essential  to  the  practical  development  of  the  survey-claim. 

39.  You  will  give  in  detail  the  value  of  each  mining  improvement  in- 
cluded in  your  estimate  of  expenditure,  and  when  a tunnel  or  other 
improvement  has  been  made  for  the  development  of  other  claims  in  con- 
nection with  the  one  for  which  survey  is  made,  your  report  must  give  the 
name,  ownership  and  survey-number,  if  any,  of  each  claim  to  which  a 
proportion  or  interest  is  credited,  and  the  value  of  the  proportion  or 
interest  credited  to  each.  The  value  of  improvements  made  upon  other 
locations  or  by  a former  locator  who  has  abandoned  his  claim  cannot  be 
included  in  your  estimate. 

40.  In  making  out  your  certificate  of  the  value  of  the  improvements, 
you  will  follow  the  form  prescribed  in  the  specimen  field-notes. 

41.  Following  your  certificate  you  will  locate  and  describe  all  other 
improvements  made  by  the  claimant  or  other  parties  within  the  bounda- 
ries of  the  survey. 

42.  If  the  value  of  the  labor  and  improvements  upon  a mineral  claim 


APPENDIX  B. 


595 


is  less  than  five  hundred  dollars  at  the  time  of  survey,  you  are  authorized 
to  file  your  affidavit  of  five  hundred  dollars  expenditure  at  any  time 
before  the  expiration  of  the  sixty  days  of  publication,  but  not  afterwards 
unless  by  special  instructions. 

DESCRIPTIVE  REPORTS  ON  PLACER-CLAIMS. 

43.  By  General  Land  Office  circular,  approved  September  23,  1882,  you 
are  required  to  make  a full  examination  of  all  placer-claims  at  the  time 
of  survey,  and  file  with  your  field-notes  a descriptive  report  in  which  you 
will  describe — 

{a)  The  quality  and  composition  of  the  soil,  and  the  kind  and  amount 
of  timber,  and  other  vegetation. 

{b)  The  locus  and  size  of  streams,  and  such  other  matters  as  may  ap- 
pear upon  the  surface  of  the  claims. 

(0  The  character  and  extent  of  all  surface  and  underground  workings, 
whether  placer  or  lode,  for  mining  purposes. 

{d)  The  proximity  of  centres  of  trade  or  residence. 

{e)  The  proximity  of  well-known  systems  of  lode  deposits  or  of  indi- 
vidual lodes. 

(/’)  The  use  or  adaptability  of  the  claim  for  placer-mining,  and 
whether  water  has  been  brought  upon  it  in  sufficient  quantity  to  mine 
the  same,  or  whether  it  can  be  procured  for  that  purpose. 

(g)  What  works  or  expenditures  have  been  made  by  the  claimant  or 
his  grantors  for  the  development  of  the  claim,  and  their  situation  and 
location  with  respect  to  the  same  as  applied  for. 

{k)  The  true  situation  of  all  mines,  salt-licks,  salt-springs,  and  mill- 
seats,  which  come  to  your  knowledge,  or  report  that  none  exist  on  the 
claim,  as  the  facts  may  warrant. 

(/)  Said  report  must  be  made  under  oath,  and  duly  corroborated  by 
one  or  more  disinterested  persons. 

44.  Descriptive  reports  upon  placer-claims  taken  by  legal  subdivisions 
are  authorized  only  by  special  order,  and  must  contain  a description  of 
the  claim  in  addition  to  the  foregoing  requirements. 

PRELIMINARY  PLAT. 

45.  You  will  file  with  your  field-notes  a preliminary  plat  on  drawing- 
paper  or  tracing-muslin,  protracted  on  a scale  of  two  hundred  feet  to  an 
inch,  on  which  you  will  note  accurately  all  the  topographical  features 
and  details  of  the  survey  in  conformity  with  the  specimen  plat  herewith. 
Pencil  sketches  will  not  be  accepted. 

REPORT. 

46.  You  will  also  submit  with  your  return  of  survey  a report  upon  the 
following  matters  incident  to  the  survey,  but  not  required  to  be  embraced 
in  the  field-notes. 

47.  If  the  meridian  from  which  your  courses  were  deflected  was  estab- 
lished by  other  means  than  by  the  solar  apparatus  attached  to  your 
transit,  you  will  state  in  detail  your  observations  and  calculations  for  the 
establishment  of  such  meridian. 


596 


SURVEYING. 


48.  If  any  of  the  lines  of  the  survey  were  determined  by  triangulation 
or  traverse,  you  will  give  in  full  detail  the  calculations  whereby  you  ar- 
rived at  the  results  reported  in  your  field-notes.  You  will  also  submit 
your  calculations  of  areas  of  placer  and  mill-site  claims  or  other  irregular 
tracts. 

49.  You  will  mention  in  your  report  the  discovery  of  any  material 
errors  in  prior  official  surveys,  giving  the  extent  of  the  same. 

ERRORS. 

50.  Whenever  a survey  has  been  reported  in  error,  the  deputy-sur- 
veyor who  made  it  will  be  required  to  promptly  make  a thorough  exami- 
nation, upon  the  premises,  and  report  the  result  under  oath  to  this  office. 
In  case  he  finds  his  survey  in  error,  he  will  report  in  detail  all  discrep- 
ancies with  the  original  survey,  and  submit  any  explanation  he  may  have 
to  offer  as  to  the  cause.  If,  on  the  contrary,  he  should  report  his  survey 
correct,  a joint  survey  will  be  ordered  to  settle  the  differences  with  the 
surveyor  who  reported  the  error. 

JOINT  SURVEY. 

51.  A joint  survey  must  be  made  within  ten  days  after  the  date  of 
order,  unless  satisfactory  reasons  are  submitted,  under  oath,  for  a post- 
ponement. 

52.  The  field-work  must  in  every  sense  of  the  term  be  a joint  and  not 
a separate  survey,  and  the  observations  and  measurements  taken  with  the 
same  instrument  and  chain,  previously  tested  and  agreed  upon. 

53.  The  deputy-surveyor  found  in  error,  or  if  both  are  in  erhor,  the 
one  who  reported  the  same  will  make  out  the  field-notes  of  the  joint  sur- 
vey, which,  after  being  duly  signed  and  sworn  to  by  both  parties,  must 
be  transmitted  to  this  office. 

54.  The  surveyor  found  in  error  will  be  required  to  pay  all  expenses 
of  the  joint  survey  and  preliminary  examinations  incident  thereto,  includ- 
ing ten  dollars  per  day  to  the  surveyor  whose  work  is  proved  to  be  sub- 
stantially correct. 

55.  Your  field-work  must  be  accurately  and  properly  performed,  and 
your  returns  made  in  conformity  with  the  foregoing  instructions.  Errors 
in  the  survey  must  be  corrected  at  your  own  expense,  and  if  the  time  re- 
quired in  the  examination  of  your  returns  is  increased  by  reason  of  your 
neglect  or  carelessness  you  will  be  required  to  make  an  additional  deposit 
for  office  work.  You  will  be  held  to  a strict  accountability  for  the  faith- 
ful discharge  of  your  duties,  and  will  be  required  to  observe  fully  the  re- 
quirements and  regulations  in  force  as  to  making  mineral  surveys.  If 
found  incompetent  as  a surveyor, careless  in  thedischarge  of  your  duties, 
or  guilty  of  a violation  of  said  regulations,  your  appointment  will  be 
promptly  revoked. 

56.  All  former  instructions  inconsistent  with  the  foregoing  are  hereby 
recalled. 


APPENDIX. 


597 


SPECIMEN  PRELIMINARY  PLAT. 


SPECIMEN  FIELD-NOTES. 


Survey  No.  4225  A and  B. 
District  No.  3. 

FIELD-NOTES. 

Of  the  survey  of  the  claim  of  The  Argentum  Mining  Company,  upon 
the  Silver  King  and  Gold  Queen  lodes,  and  Silver  King  Mill  site,  in 
Alpine  Mining  District,  Lake  county,  Colorado. 

Surveyed  by  George  Lighifoot,  U.  S.  Deputy  Mineral  Surveyor. 
.Survey  begun  April  22d,  1886,  and  completed  April  24111,  1886. 

Address  of  claimant,  Wabasso,  Colorado. 


FEET. 


1242. 

1365.28 


152. 

300- 


SURVEY  NO.  4225  A. 

SILVER  KING  LODE. 

Beginning  at  Cor.  No.  i. 

Identical  with  Cor.  No.  i of  the  location. 

A spruce  post,  5 ft.  long,  4 ins.  square,  set  2 ft.  in  the 
ground,  with  mound  of  stone,  marked  whence 

The  W.  1 cor.  Sec.  22,  T.  1 1 S.,  R.  81  W.  of  the  6th  Prin- 
, cipal  Meridian,  bears  S.  79°  34'  W.  1378.2  ft. 

Cor.  No.  i,Gottenburg  lode  funsurveyed),  Neals  Mattson, 
claimant,  bears  S.  40°  29'  W.  187.67  ft. 

A pine  12  ins.  dia.,  blazed  and  marked  B.  T.  A,  bears 
S.  7°  25'  E.  22  ft. 

Mount  Ouray  bears  N.  11°  E. 

Hiawatha  Peak  bears  N.  47°  45'  W. 

Thence  S.  24^"  45'  W. 

Va.  15°  12'  E. 

To  trail,  course  N.  W.  and  S.  E. 

To  Cor.  No.  2. 

A granite  stone  25x9x6  ins.,  set  18  ins.  in  the  ground, 
chiselled  4/^  A,  whence 

Cor.  No.  2 of  the  location  bears  S.  24°  45'  W.  134.72  ft. 
Cor.  No.  I,  Sur.  No.  2560,  Carnarvon  lode,  David  Davies  et 
al.,  claimants  bears  S.  3°  28'  E.  116. 6 ft. 

North  end  of  bridge  over  Columbine  creek  bears  S.  65®  1 5' 
E.  650  ft. 

Thence  N.  65®  15'  W. 

Va.  15®  20'  E. 

Intersect  line  4-1,  Sur.  No.  2560,  at  N.  38®  52'  W.,  231.2  ft. 

from  Cor.  No.  i. 

To  Cor.  No.  3. 


APPENDIX  B. 


599 


FEET. 

A cross  at  corner-point,  and  A chiselled  on  a granite 

rock  in  place,  20 x 14 x 6 ft.  above  the  general  level,  whence 
Cor.  No.  3 of  the  location  bears  S.  24°  45'  W.  134.72  ft. 

734 

A spruce  16  ins.  dia.,  blazed  and  marked  B.  T.  A, 

bears  S,  58°  W.  18  ft. 

Thence  N.  24°  45'  E. 

Va.  15°  20'  E. 

Intersect  line  4-1  Sur.  No.  2560  at  N.  38°  52'  W.  396.4  ft. 
from  Cor.  No.  i. 

150. 

237. 

1000.9 

Intersect  line  6-7  of  this  survey. 

To  trail,  course  N.  W.  and  S.  E. 

Intersect  line  2-3,  Gottenburg  lode,  at  N.  25°  56'  W.  76.26  ft. 
from  Cor.  No.  2. 

1365.28 

To  Cor.  No.  4. 

Identical  with  Cor.  No.  4 of  the  location. 

A pine  post  4.5  ft.  long  5 ins.  square,  set  one  foot  in  the 
ground,  with  mound  of  earth  and  stone,  marked 
whence 

28.5 

A cross  chiselled  on  rock  in  place,  marked  B.  R.  A, 

bears  N.  28°  10'  E.  58.9  ft. 

Thence  S.  65°  15'  E. 

Va.  15°  12'  E. 

Intersect  line  4-1,  Gottenburg  lode,  at  N.  25°  56'  W 285.15  ft. 
from  Cor.  No.  i. 

65. 

300. 

Intersect  line  5-6  of  this  survey. 

To  Cor.  No.  I,  the  place  of  beginning. 

285. 

315- 

GOLD  QUEEN  LODE. 

Beginning  at  Cor.  No.  5, 

A pine  post  5 ft.  long,  5 ins.  square,  set  2 ft.  in  the  ground, 
with  mound  of  earth  and  stone,  marked  A,  whence 

Cor.  No.  I of  this  survey  bears  S.  14°  54'  E.  370.16  ft. 

A pine  18  in.  dia.  bears  S.  33°  15'  W.  51  ft.,  and  a silver 
spruce  13  ins.  dia.  bears  N.  60°  W.  23  ft.,  both  blazed  and 
marked  B.  T.  A. 

Thence  S.  24°  30'  W. 

Va.  15°  14'  E. 

Intersect  line  4-1  of  this  survey. 

Intersect  line  4-1.  Gottenburg  lode,  at  N.  25°  56' W.  237.78  ft. 
from  Cor.  No.  i. 

688.3 

Intersect  line  1-2,  Gottenburg  lode,  at  N.  64^04'  E.  12.23 
from  Cor.  No.  2. 

1438. 

1500. 

To  trail,  course  N.  W.  and  S.  E. 

To  cor.  No.  6, 

s 

! 

A granite  stone  34  x 14x6  ins.,  set  one  foot  in  the  ground 
to  bedrock,  with  mound  of  stone,  chiselled  j/g-j  A,  whence 

A cross  chiselled  on  ledge  of  rock  marked  B.  R. 
bears  due  north  12  ft. 

6oo 


SURVEYING. 


FEET. 

70.3 

223.37 

300. 


38.43 

165. 

1043.73 

1432.90 

1500. 


300. 


Thence  N.  65°  30'  W. 

Va.  15°  20'  E. 

Intersect  line  3-4  of  this  survey. 

Intersect  line  4-1,  Sur.  No.  2560  at  N.  38°  52'  W.  567.28  ft. 

from  Cor.  No.  i. 

To  Cor.  No.  7. 

A cross  at  corner-point  and^’g-y  A chislled  on  a granite 
boulder  12  x6  x 3 ft.  above  ground,  whence 

A cross  chiselled  on  vertical  face  of  cliff,  marked  H.  R. 
^^27  hears  N.  72°  W.  56.2  ft. 

A pine  14  ins.  dia.,  blazed  and  marked  B.  T.  hears 

N.  10°  E.  39  ft. 

Thence  N 24®  30'  E. 

Va.  not  determinerl  on  account  of  local  attraction. 
Intersect  line  4-1,  Sur.  No.  2560,  at  N.  38®  52' W.  653  ft.  from 
Cor.  No.  I. 

To  trail,  course  N.  W.  and  S.  E. 

Intersect  line  2-3,  Gottenburg  lode,  at  N.  25°  56'  W.  379.06  ft. 
from  Cor.  No.  2. 

Intersect  line  4-1,  Gottenburg  lode,  at  N.  25®  56' W.  626.94  ft. 

from  Cor.  No.  i. 

To  Cor.  No.  8. 

A spruce  post  6 ft.  long,  5 ins.  square,  set  2.5  ft.  in  the 
ground  with  mound  of  stone,  marked  A,  whence 
A cross  chislled  on  rock  in  place,  marked  B.  R. 
bears  S.  9°  12'  E.  1 5.8  ft. 

A pine,  20  ins.  dia.,  blazed  and  marked  B.  T.  hears 

N.  83®  E.28.5  ft. 

Thence  S.  65°  30'  E. 

Va.  15®  16' E. 

To  Cor.  No.  5,  the  place  of  beginning. 


Area. 

Total  area  of  Silver  King  lode 9-403  acres 

Less  area  in  conflict  with 

Sur.  No.  2560 124  acre 

Gottenburg  lode  1*363  “ 1.487  acres 


Net  area  of  Silver  King  lode 7*9i6  acres 


Total  area  of  Gold  Queen  lode 10.331  acres 

Area  in  conflict  with 

Sur.  No.  2560 034  “ 

Gottenburg  lode 2.679  “ 

Silver  King  lode 1.887  “ 


Silver  King  lode  (exclusive  of  conflict  of 
said  Silver  King  lode  with  the  Gottenburg  lode)  1.309 


APPENDIX  B. 


6oi 


FEET. 


90. 

208. 

504.8 


351- 

3944 


15. 

40. 


370. 

647.2 


Total  area  of  Gold  Queen  lode 10.331  acres 

Less  area  in  conflict  with 

Sur.  No.  2560 034  acre 

Gottenburg  lode 2.679  “ 

Silver  King  lode i-309  “ 4- 022  acres 


Net  area  of  Gold  Queen  lode 6.  309  acres 

“ “ Silver  King  lode 7-9i6 

Net  area  of  lode  claim 14.225  acres 


SURVEY  NO.  4225  B. 

SILVER  KING  MILL-SITE. 

Beginning  at  Cor.  No.  i, 

A gneiss  stone  32x8x6  ins.,  set  2 ft.  in  the  ground,  chis- 
elled B,  whence  W.  4 cor.  Sec.  22,  T.  ii  S,  R.  81  W.  of 
the  6th  Principal  Meridian,  bears  N.  80°  W.  1880  ft. 

Cor.  No.  I,  Sur.  No.  4225  A,  bears  N.  40°  44'  W.  760.2  ft. 
A cottonwood  18  ins.  dia.,  blazed  and  marked  4^Vt 
bears  S.  5°  30'  E.  17  ft. 

Thence  S.  34°  E. 

Road  to  Wabasso,  course  N.  E.  and  S.  W. 

Right  bank  of  Columbine  creek,  75  ft.  wide,  flows  S.  W. 

To  Cor.  No.  2, 

An  iron  bolt  18  ins.  long,  i in.  dia.,  set  one  foot  in  rock  in 
place,  chiselled  4^3-  B,  whence 

A cottonwood,  blazed  and  marked  B.  T.  bears  E. 

182  ft. 

Thence  S.  56°  W. 

Left  bank  of  Columbine  creek. 

To  Cor.  No.  3, 

A point  in  bed  of  creek,  unsuitable  for  the  establishment 
of  a permanent  corner. 

Thence  N.  34°  W. 

Right  bank  of  Columbine  creek. 

To  witness-corner  to  Cor.  No.  3, 

A pine  post  4.5  ft.  long,  5 ins.  square,  set  one  foot  in 
ground,  with  mound  of  stone,  marked  W.  C.  4^25  whence 
A cottonwood  15  ins.  dia.  bears  N.  ii"  E.  16.5  ft.  and  a 
cottonwood  19  ins.  dia.  bears  N.  83°  W.  23  ft.,  both  blazed 
and  marked  B.  T.  W.  C.  B. 

Road  to  Wabasso,  course  N.  E.  and  S.  W. 

To  Cor.  No.  4, 

A gneiss  stone  24 x 10x4  ins.,  set  18  ins.  in  the  ground, 
chiselled  72W  whence 

A cross  chiselled  on  ledge  of  rock,  marked  B.  R.  B, 
bears  N.  85°  10'  E.  26.4  ft. 

Thence  N.  48^43'  E. 


6o2 


SURVEYING. 


FEET.  I 

125.5  I To  Cor.  No.  5, 

I A gneiss  stone  30x8x5  ins.,  set  2 ft.  in  the  grounrl, 
I cliisellecl  13. 

Thence  S.  34°  E. 

1 58.3  To  Cor.  No.  6, 

A pine  post  5 ft.  long,  5 ins.  square,  set  2 ft.  in  the  ground 
with  mound  of  earth  and  stone,  marked  whence 

A pine  12  ins.  dia.,  blazed  and  marked  I>.  13,  bears 

S.  33“  E.  63.5  ft. 

Thence  N.  56°  E. 

270.  To  Cor.  No.  I,  the  place  of  beginning. 

Containing  5 acres. 

Variation  at  all  the  corners,  15°  20'  E. 


The  surveys  of  the  Gold  Queen  lode  and  Silver  King  mill 
site  are  identical  with  the  respective  locations. 


LOCATION. 

This  claim  is  located  in  the  \V.  ^ Sec.  22,  T.  1 1 S.,  R.  81  W. 


EXPENDITURE  OF  FIVE  HUNDRED  DOLLAR.S. 

I certify  that  the  value  of  the  labor  and  improvements 
upon  this  claim,  placed  thereon  by  the  claimant  and  its 
grantors,  is  not  less  than  five  hundred  dollars,  and  that  said 
improvements  consist  of 

The  discovery  shaft  of  the  Silver  King  lode,  6x3  ft.,  10 
ft.  deep  in  earth  and  rock,  which  bears  from  Cor.  No.  2 N.  6° 
42'  W.  287.5  ft.  Value  S80. 

An  incline  7x5  ft.,  45  ft.  deep  in  coarse  gravel  and  rock, 
timbered,  course  N.  58°  15'  W.,  dip  62°,  the  mouth  of  which 
bears  from  Cor.  No.  2 N.  i5°37'E.  908  ft.  Value  S550. 

The  discovery  shaft  of  the  Gold  Queen  lode.  5x5  ft.,  18 
ft.  deep  in  rock,  which  bears  from  Cor.  No.  7 N.  67°  39'  E. 
219.3  ft.,  at  the  bottom  of  which  is  a cross-cut  6.5x4  ft. 
running  N.  59^26'  W.  75  ft. 

Value  of  shaft  and  cross-cut,  $1,000. 

A log  shaft-house  14  ft.  square,  over  the  last-mentioned 
shaft.  Value  $100. 

Two-thirds  interest  in  an  adit  6.5  x 5 ft.,  running  due  west 
835  ft.,  timbered,  the  mouth  of  which  bears  from  Cor.  No.  2 
N.  61°  15'  E.  920  ft. 

This  adit  is  in  course  of  construction  for  the  development 
of  the  Silver  King  and  Gold  Queen  lodes  of  this  claim,  and 
Sur.  No.  2560,  Carnarvon  lode,  David  Davies  et  al.,  claimants, 
the  remaining  one-third  interest  therein  having  already  been 
included  in  the  estimate  of  five  hundred  dollars  expenditure 
upon  the  latter  claim,  Total  value  of  adit,  $13,000. 


APPENDIX  B. 


603 


A drift  6.5  x4  ft.  on  the  Silver  King  lode,  beginning  at  a 
point  in  adit  800  ft.  from  the  mouth,  and  running  N.  20°  20' 
E.  195  ft.,  thence  N,  54°  15'  E.  40  ft.  to  breast.  Value  $2,800. 

I further  certify  tliat  no  portion  of  the  improvements 
claimed  have  been  included  in  the  estimate  of  five  hundred 
dollars  expenditure  upon  any  other  claim. 


OTHER  IMPROVEMENTS. 

A log  cabin  35x28  ft.,  the  S.  W.  corner  of  which  bears 
from  Cor.  No.  7 N.  30°  44'  E.  496  ft. 

A dam  4 ft.  high,  50  ft.  long,  across  Columbine  creek,  the 
south  end  of  which  bears  from  Cor.  No.  2 of  the  mill-site  N. 
58°  20'  W.  240  ft. 

Said  cabin  and  dam  belong  to  The  Argentum  Mining 
Company. 

An  adit  6x4  ft.,  running  N.  70°  50'  W.  100  ft.,  the  mouth 
of  which  bears  from  Cor.  No.  5 S.  58°  12'  W.  323  ft.  belong- 
ing to  Neals  Mattson,  claimant  of  the  Gottenburg  lode. 


INSTRUMENT. 

The  survey  was  made  with  a Young  & Sons  mountain 
transit  No.  5322,  with  Smith’s  solar  attachment.  The  courses 
were  deflected  from  the  true  meridian  as  determined  by  solar 
observations.  The  distances  were  measured  with  a 50  ft. 
steel  tape. 

employe’s  certificate. 

List  of  the  names  of  individuals  employed  to  assist  in  running,  meas- 
uring and  marking  the  lines  and  corners  described  in  the  foregoing  field- 
notes  of  the  survey  of  the  claim  of  The  Argentum  Mining  Company 
upon  the  Silver  King  and  Gold  Queen  lodes  and  Silver  King  mill-site,  in 
Alpine  Mining  District,  Lake  County,  Colorado. 

William  Sharp, 
Robert  Talc. 

We  hereby  certify  that  we  assisted  George  Lightfoot  U.  S.  Deputy 
Mineral  Surveyor,  in  surveying  the  exterior  boundaries  and  marking  the 
corners  of  the  claim  of  The  Argentum  Mining  Company  upon  the  Silver 
King  and  Gold  Queen  lodes  and  Silver  King  mill-site  in  Alpine  Mining 
District,  Lake  County,  Colorado,  and  that  said  survey  has  been  in  all  re- 
spects, to  the  best  of  our  knowledge  and  belief,  well  and  faithfully  sur- 
veyed and  the  boundary  monuments  planted  according  to  the  instruc- 
tions furnished  by  the  Surveyor-General. 

William  Sharp, 
Robert  Talc. 

Subscribed  and  sworn  to  by  the  above-named  persons  before  m-e,  this 
26th  day  of  April,  1886. 

[Seal] 


John  Doolittle, 

Notary  Public. 


6o4 


VE  Y INC. 


surveyor’s  oath. 

I,  George  Lightfoot,  U.  S.  Deputy  Mineral  Surveyor,  do  solemnly 
swear  that  in  pursuance  of  an  order  from  Jas.  A.  Dawson,  Surveyor-Gen- 
eral of  the  public  lands  in  the  State  of  Colorado,  bearing  date  the  30th 
day  of  March  1886,  and  in  strict  conformity  with  the  laws  of  the  United 
States,  and  instructions  furnished  by  said  Suiweyor-General,  I have 
faithfully  surveyed  the  claim  of  The  Argentum  Mining  Company  upon 
the  Silver  King  and  Gold  Queen  lodes  and  Silver  King  mill-site  in  Alpine 
Mining  District,  Lake  County,  Colorado,  and  do  further  solemnly  swear 
that  the  foregoing  are  the  true  and  original  field-notes  of  such  survey, 
and  that  the  improvements  are  as  therein  stated. 

George  Lightfoot, 

U.  S.  Deputy  M incral  Surveyor. 

Subscribed  by  said  George  Lightfoot,  U.  S.  Deputy  Mineral  Surveyor, 
and  sworn  to  before  me  this  26th  day  of  April,  1886. 

[Seal]  John  Doolittle, 

Notary  Public. 


APPENDIX  C 


FINITE  DIFFERENCES. 


THE  CONSTRUCTION  OF  TABLES. 

In  the  accompanying  figure  the  ordinates  are  spaced  at  the  uniform 
distance  / apart.  Let  the  successive  values  of  these  ordinates,  and  their 
several  orders  of  differences,  be  represented  by  the  following  notation: 


Values  of  the  function,  hi,  hi,  hz,  kz,  kz. 

First  order  of  differences,  A' A' A' A' h^,  A'h^,  A'^^. 

Second  “ “ A"Ao,  A"  k^,  A"h^,  A"  h^,  A"h^. 

Third  “ “ ' A’''hz.  A'"a„  A'"/^,. 

Fourth  “ “ 


etc., 


etc. 


6o6 


SCrji^  VE  YING. 


We  may  now  write 

hi  ■=.  ^'Aq’,  1 

hi  = hi  -j-  z/'/;,  = ho  + z/'Ao  + ^'Ao  + ^"Ao  = ^^0  + 2^'//o  + ^"Ao'. 

h%  — hi  -f-  A' = ho  + 3'^Vi’o  4"  3-^’V/o  + Ao\ 
h\  = ho  -j-  4‘^'Ao  4“  ^^  "Ao  4“  4-^  "ho  4“  ^^''ho\  * 

hn  = /^o  4-  ho  4 T72~^  VT^~3 ^ ^0  4-  etc. 


(0 


It  is  to  be  observed  that  the  coefTicients  follow  the  law  of  the  bino- 
mial development.  It  is  also  seen  that  the  first  of  the  successive  orders 
of  differences  are  alone  sufficient  to  enable  any  term  of  the  function  to 
be  computed.  We  will  now  proceed  to  find  these  first  terms  of  the 
several  orders  of  differences  for  any  given  equation. 

Almost  all  functions  of  a single  variable  can  be  developed  by  the  aid 
of  Maclaurin’s  Formula,  in  the  form 


y'o  — Co  CiXo  -(-  CiXo"^  4“  ^3-^0^  4“  CiiXo^  -f-  etc (2) 

If  X take  an  increment  A^,  thus  becoming  Xi,  the  cha^ige  in  yo  will 

be  represented  by  A' and  its  value  will  be  the  new  value  of  the  function 
minus  its  initial  value,  or  A'y^  =yi  —yo.  By  putting  x + Aj^  for  x in  the 
above  equation,  developing,  subtracting  the  original  equation,  and  re- 
ducing, we  would  obtain 

— yo  = ^Vo  = (^1  4-  2CiXo  4"  sCsXo^  4"  4 

4“  (C2  4"  3^3^0  4“  ^C4Xo^)A'^^  -j-  (C3  -f-  4C4Xo)A^x  CiA^x  , • (3) 

assuming  that  the  function  stops  with  C4X0*. 

If  jfi  should  now  take  another  increment  equal  to  the  previous 

one,  we  would  have  Xi  — Xi  + z/^  and  yi  = yi  4-  A'y^_  Now  A' is  the 

value  A'y  when  Xo  has  become  Xu  and  the  difference  between  A' and 
A'y^  is  the  change  in  the  value  of  A' y^  due  to  this  change  in  .r. 

Hence  z/'ji  — = 4/'Vo- 

To  find  the  value  of  substitute  x Ax  for  Jir  in  equation  (3), 
develop,  subtract  equation  (3),  reduce,  and  obtain 

A" y^  — (2C2  -f-  6C3X0  -f-  I2C4.Yo*)zf'^^  T"  (6C3  4“  24^^4>^o)zf®^ (4) 


Similarly  we  find 

z/''Vo  = (6a  4-  24a^o)^3^4-  3(>C4^^x (5) 

A^'^y^  = 24az/'‘^(a  constant) (6) 


APPENDIX  C. 


607 


From  the  above  development  we  see — 

1.  Til  at  ihe  iiianber  of  orders  of  differences  is  equal  to  the  highest  ex- 
ponent of  the  variable  involved,  the  last  difference  being  a constant. 

2.  That  if  nny  initial  value  of  the  variable  be  taken,  the  first  of 
the  several  orders  of  differences  can  be  obtained  in  terms  of  this  initial 
value,  its  constant  increment,  and  the  constant  coefficients.  This  fur- 
nishes a ready  means  of  computing  a table  of  values  of  the  function,  if 
it  can  be  represented  in  the  form  of  equation  (i).  Evidently  if  the  ini- 
tial value  of  the  variable  (xo)  be  taken  as  zero,  the  evaluation  for  the 
several  initial  differences  is  much  simplified,  for  then  all  the  terms  in  x 
disappear.  If  the  constant  increment  be  also  taken  as  unity,  the  labor 
is  still  further  reduced. 

Example. — Construct  a table  of  values  of  the  function 

/ = 50  — 40X  -|-  20x^  -|-  4x'^  — (7) 

Let  the  initial  value  of  the  variable  be  zero  and  the  increments  unity. 

Evaluating  the  initial  differences  by  equations  (3)  to  (6),  we  find,  for  a-q  = o, 
and  = i, 


yo  = 

-j-  50; 

Vo2  — Cl  -j-  Ca  -j-  C3  -{-  C4 

= - 17; 

A'  j'q  = iCi  -j-  6(73  ~1“  14C4 

= + 50; 

A"'yO  — 6C3  — j— 

= — 12; 

A'^'^yo  = 24  C4 

= - 24. 

From  these  initial  values  we  may  readily  construct  the  following 
table : 


Values  of 

X. 

Values  of 

ist  Differences. 

AV 

2d  Differences. 
A-,. 

3d  Differences. 
A-V. 

4th  Differences. 

Aivy. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

etc. 

50 

33 

66 

137 

210 

225 

98 

— 279 

— 1038 

etc. 

- 17 

+ 33 

+ 71 

+ 73 

+ 15 

- 127 

- 377 

- 759 

etc. 

+ 50 
+ 38 
+ 2 

— 58 

— 142 

— 250 

— 382 

etc. 

— 12 

— 36 

— 60 

— 84 

— 108 

— 132 

etc. 

.-  24 

- 24 

- 24 

- 24 

- 24 

etc. 

* Fig.  152  is  the  locus  of  this  curve,  the  ordinates  being  taken  from  th^.s 
column. 


6o8 


SUR  VE  YING. 


The  initial  values  in  all  the  columns  beiii,"  ^iven,  the  table  is  made 
by  continual  additions,  one  column  after  anollier,  workinj:!^  from  right  to 
left.  Thus,  the  4th  difTerence  being  constant,  the  initial  value,  — 24,  is 
simply  repeated  indefinitely.  The  column  of  3d  clifTcrcnccs  is  now  com-, 
puted  by  adding  continuously  —24  to  the  preceding  value.  The  column 
of  2d  differences  is  next  made  out,  the  quantity  to  be  added  each  time 
being  the  intervening  3d  difference,  which  is  not  constant.  In  a similar 
manner  proceed  with  the  column  of  ist  differences,  and  finally  with  the 
values  of  the  function  itself. 

The  above  formulae  apply  to  all  functions  of  a single  variable  not 
higher  than  the  fourth  degree.  Evidently  any  of  the  C’ coefficients  may 
be  zero,  and  so  cause  one  or  more  of  the  powers  of  .r  to  entirely  disappear. 
If  the  variable  is  involved  to  a higher  degree  than  the  fourth,  a new  de- 
velopment may  be  made,  or  the  initial  values  of  the  successive  orders  of 
differences  may  be  determined  by  simply  evaluating  the  function  for  a 
series  of  successive  values  of  the  variable,  one  more  in  number  than  the 
degree  of  the  equation,  and  then  working  out  the  successive  columns  of 
differences  from  these  until  the  last,  or  constant,  difference  is  found. 
The  table  may  then  be  continued  by  combining  these  differences,  as  be- 
fore. Thus  in  the  above  example  the  first  five  values  of  j/  might  have 
been  found  by  direct  evaluation  of  the  function  for  the  corresponding 
values  of  x,  and  then  the  successive  differences  taken  out  until  the  con- 
stant fourth  difference,  — 24,  was  found.  This  can  always  be  done  with- 
out resorting  to  any  algebraic  discussion  as  given  above. 


THE  EVALUATION  OF  IRREGULAR  AREAS. 

The  ordinates  to  any  curve,  as  that  in  Fig.  152  for  instance,  may  be 
represented  by  such  an  equation  as  the  last  of  equations,  (i ),  where  the 
length  of  any  ordinate  is  given  in  terms  of  its  number  from  the  initial  or- 
dinate, the  value  of  this  first  ordinate,  and  the  first  of  the  successive 
orders  of  differences.  This  equation  is 


, , , ~ i) 

hn  = /lo  /iq  -j- ^ /i’o 


n {n  — i)  {it  — 2) 


1.2.3 


^'"y^o-f-etc., 


where  h„  is  the  ;zth,  and  therefore  any  ordinate  to  the  curve.  The  con- 
stant distance  between  the  ordinates  apparently  does  not  enter  the  equa- 
tion, but  it  is  really  represented  in  the  several  /I’s. 

By  the  calculus  the  area  of  any  figure  included  between  any  curve, 


the  axis  of  abscissas,  and  two  extreme  ordinates  is  A 


-j: 


hdx,  where  h 


is  the  general  value  of  an  ordinate,  = hn  in  the  above  equation,  where  it 
is  shown  to  be  a function  of  n.  Also  x = nl  where  / is  the  constant 
distance  between  ordinates,  whence  dx  = ld7i. 
ues  of  /2  and  dx,  we  have 


Substituting  these  val- 


APPENDIX  C. 


609 


0 


Integrating  this  equation,  we  obtain 


From  the  schedule  of  differences  on  p.  605  we  may  at  once  find  the 
initial  values  of  the  several  orders  of  differences  in  terms  of  the  succes- 
sive values  of  the  function.  Thus 

A'Ao  = hx  — h^\ 

A" — A' — A' — hi  — 2h\ 

A'"/to  = + A'/,^  - hi  — 2,hi  + 2>hi  - ho; 

AKho  = A"'h^  — A'" ho  = A”h.,  — 2A"h^  -f  A" ho  = A’ ho  — sA'h^  + 3^'Aj  —A'ho 

— hi  — 4'^3  6^2  — 4^1 

Again,  the  coefficients  follow  the  law  of  the  binomial  development, 
and  we  may  write 


n{n  — i){n  — 2) 
1.2.3 


/^«-3  -(-  etc.  . (10) 


By  the  aid  of  this  equation  we  may  now  substitute  for  the  several 
initial  differences  in  equation  (9)  their  values  in  terms  of  the  successive 
values  of  the  function.  Also  for  any  area  divided  into  n sections  by  or- 
dinates, uniformly  spaced  a distance  / apart,  equation  (9)  will  give  the 
area  in  terms  of  /,  71,  and  the  several  ordinates,  when  these  latter  are  sub- 
stituted for  the  z/’s  by  means  of  eq.  (10). 

Thus,  for  n = \,  equation  (9)  becomes 


(II) 


39 


6io 


SURVEYING. 


which  is  the  Trapezoidal  Rule. 

Fer  n = 2, 

^ =/(2/i.  + 24'^. + (5- = /(i/, 0+  3/,,+ i/;,)  = ^(//.+  4/S, + //,),  (12) 


which  is  called  Simpson  s \ Rule. 

If  r — 'll  = total  length  of  figure,  this  formula  becomes 

/' 

^ (■^^0  -j-  4^0  -j-  hi), (l2fl) 


which  is  the  well-known  form  of  the  Prismoidal  T'ormula,  and  it  would 
be  that  formula  if  areas  were  substituted  for  ordinates. 

If  n = 3, 


^ ~ H"  3-^^!  3hi  + hi), (13) 

which  is  called  Simpson  s f Rule. 

If  n = 4, 


A — - [7  (^0  -f-  hi)  32  {Ji\  -f-  hi)  -j-  12,^2]. 

45 


• . (14) 


If  n = 6, 

A = l[6ho  iSA'/i^  -j-  2yA"/tQ  is,A"' Jiq  + 

If  now  the  coefficient  of  be  changed  from  to  which 

would  not  affect  curves  of  a degree  less  than  the  sixth,  the  resulting 
equation,  when  the  //s  are  substituted  for  the  A’s,  takes  the  following 
very  simple  form  ; 

A = ~ [//o  + h-2  hi  -{-  liQ  S {hi  -{-  -^3  + hi)  -|-  hi\,  . . (15) 

which  is  called  IVeddel’s  Ride. 

For  a greater  number  of  ordinates  than  seven,  it  is  best  to  use  either 
equation  (12),  (13),  or  (15)  several  times,  as  the  formulae  become  very 
complicated  for  7t  > 6. 


APPENDIX  D. 


DERIVATION  OF  FORMULiE  FOR  COMPUTING  GEOGRAPH- 
ICAL COORDINATES  AND  FOR  THE  PRO- 
JECTION OF  MAPS* 


Let  Fig.  153  represent  a distorted  meridian  section  of  the  earth. 
Let  a = the  major  and  b the  minor  semi-axes. 


Then 


= the  ellipticity. 


The  eccentricity  is  given  by 

— IP'  o 

= 5 — , whence  \ 

The  line  nm  = A"  is  the  normal  to  the  curve  at  n ; 
the  angle  ncd  = A is  the  geocentric  latitude  ; 
while  nld  = Z is  the  geodetic  latitude. 


The  geodetic  latitude  is  always  understood,  ^s  it  is  the  latitude  ob- 
tained from  astronomical  observations. 

It  is  desirable  to  find  the  length  of  the  line  nl,  of  the  normal  nm,  and 
of  the  radius  of  curvature /V',  all  in  terms  of  e,  L,  and  a.  Also  to  find 
the  geocentric  latitude  in  terms  of  a,  b,  and  L. 

To  find  nl,  we  have 


nl—^/  nd'  -f  dl^=  |/ ^ 


For  the  ellipse, 


whence 


dy  _ b’^x ^ 
dx  al^y' 


-j-  (i  — e^yx^ 


(I) 


(2) 


* See  Chapters  XIV.  and  XV.  for  the  use  of  the  formulae. 


6i2 


SUR  VE  YING. 


But  the  equation  of  the  ellipse  in  terms  of  its  eccentricity  is 


,4 


whence 


nl  — V y’^e^  -f-  ~ 


Fig.  153. 

Squaring,  remembering  thatj^  = sin  L,  we  have,  after  reducing, 



• (i  — sin^  L)i 

To  find  the  length  of  the  normal  nm  = N,  we  have 
nm  \ nl  w x\  dl. 


But 


whence 


dl  = nd  tan  dnl  ■=  y~—^—  x — — e^)x\ 

dx 


nm  = N = 


1 — (i  — sin'-'  L)i 


APPENDIX  D. 


613 


To  find  the  geocentric  latitude  in  terms  of  a,  b,  and  Z,  we  have 
A = ncd  ; L — nld. 

Since  both  have  the  common  ordinate  nd,  we  may  write 
tan  A : tan  L ::  dl  : dc. 
b^ 

But  dl  = —X  from  (4),  and  dc  = x, 

b^ 

whence  tan  A = ^ tan  Z.  . . (C) 

To  find  the  radius  of  curvature,  Z,  we  have,  in  general, 


(5) 


For  the  ellipse, 


dy 

dx 


whence 


h^x_  dy  _ b* 

d^  y dx^  ~ a-y  ’ 


(6) 


To  get  this  in  terms  of  a,  e,  and  Z,  we  have,  from  Fig.  153, 

d^  (i  — sin*'^  Z 


nl  sin®  Z = 


sin®  Z 


Also  from  the  equation  of  the  ellipse  in  terms  of  its  eccentricity  we 
have 

'I  — 2 _ _Z!__  _ — sin®  Z) 

I — ^®  I — ^®  sin®  Z * 


We  may  now  find 


d^lA 

I — ^®  sin®  Z’ 


a^y^  4"  — 


6i4 


SUA'  VE  YAXC. 


or 


(ay + ^^^5)5  = 


(i  — sin‘^  Z)3' 


(7) 


Substituting  this  in  (6),  wc  obtain 


^ ^ _ a — e'‘') 

a ’ (i  — e'^  sin''  L)l  (i  — e'^  sin''  L)^ 


. . (D) 


The  radius  of  curv^aturc  of  the  meridian,  R,  and  the  radius  of  curva- 
ture of  the  great  circle  perpendicular  to  a given  meridian  at  the  point 
where  they  intersect,  wliicl)  is  the  normal,  N,  are  the  most  important 
functions  in  geodetic  formulae.  We  will  now  derive  the  equations  used 


on  the  U.  S.  Coast  and  Geodetic  Survey  for  computing  geodetic  positions 
from  the  results  of  a primary  triangulation. 

In  Fig.  154,  let  A and  B be  two  points  on  the  surface  of  the  earth, 
which  were  used  as  adjacent  triangulation-stations.  The  distance  between 
them,  the  azimuth  of  the  line  AB  at  one  of  the  stations,  and  the  latitude 


APPENDIX  D. 


615 


and  longitude  of  one  station  are  supposed  to  be  known  ; the  latitude  and 
longitude  of  the  other  station,  and  tlie  back  azimuth  of  the  line  joining 
them,  are  to  be  found. 

Let  L'  = known  latitude  of  B ; 

Z = unknown  latitude  of  A ; 

K — known  length  of  line  AB  reduced  to  sea-level; 

s — length  of  arc  AB  — 

Z — known  azimuth  of  BA  at  Z; 

Z — unknown  azimuth  oi  A B z.\.  A\ 

M'  — known  longitude  of  B\ 

M = unknown  longitude  of  A. 

The  angle  APB  formed  by  the  two  meridional  planes  through  A and 
B is  the  difference  of  longitude  M—  M'  — AM. 

The  difference  of  latitude  is,  L — L'  — AL  — Bl  in  the  figure.  Al  is 
the  trace  of  a parallel  of  latitude  through  A and  I is  its  intersection  with 
the  meridian  through  B.  AP'  is  the  trace  of  a great  circle  through  A 
perpendicular  to  the  meridian  through  B,  and  P'  is  the  point  of  its  inter- 
section with  tliat  meridian. 

The  normals  are  B71'  = N'  and  Ati  = N.  The  radii  of  curvatvre  are 
Br  — R'  and  Ar  = R. 

The  latitude  and  longitude  of  A,  and  the  azimuth  of  the  line  AB  from 
A towards  B,  can  now  be  found  by  solving  the  spherical  triangle  APB. 
Thus  Z = 90°  - ; M ^ M'  - Al\  and  Z = 180°  - PAB. 

Although  the  line  AB  lies  on  the  surface  of  a spheroid,  if  a sphere  be 
conceived  such  that  its  surface  is  tangent  internally  to  the  surface  of  the 
spheroid  on  the  parallel  of  latitude  passing  through  the  middle  point  of 
the  line  AB,  then  this  line  will  lie  so  nearly  in  the  surface  of  the  sphere, 
that  no  appreciable  error  is  made  by  assuming  it  to  be  in  its  surface.  The 
triangle  ABP  then  becomes  a triangle  on  the  surface  of  the  tangent 
sphere,  and  hence  is  a true  spherical  triangle.  The  sphere  is  defined  by 
taking  its  radius  equal  to  the  normal  to  the  meridian  at  the  mean  lati- 
tude of  the  points  A and  B.  Since  this  mean  latitude  is  unknown,  the 
formulse  are  first  derived  for  the  latitude  of  B,  Z',  and  then  a correction 
applied  to  reduce  it  to  the  mean  latitude. 


THE  DIFFERENCE  OF  LATITUDE. 

Let  it  first  be  required  to  find  Z from  Z',  or  find  AL  = Z — Z'. 

If  we  write  /,  for  the  co-latitudes  of  Z,  Z',  and  z'  for  180°  — Z',  we 
liave,  from  the  spherical  triangle  ABP, 


cos  I = cos  I'  cos  s -}-  sin  I'  sin  s cos  z' , 


(8) 


6i6 


SUR  VE  YING. 


By  means  of  Taylor’s  Formula  we  may  find  the  value  of  / in  ascending 
powers  of  s,  and  since  s is  always  a very  small  arc  in  terms  of  the  radius, 
usually  from  20  to  60  minutes,  the  series  will  be  rapidly  converging 
By  means  of  Taylor’s  Formula,  we  may  at  once  write 


/=/'  + 


I d^-r 


I dH 


etc. 


(9) 


We  will  use  but  the  first  three  terms  of  this  development,  the  fourth 
term  being  used  only  in  the  largest  primary  triangles. 

The  derivation  of  the  successive  dilferential  coefficients  of  / with 
respect  to  s is  the  most  difficult  portion  of  this  general  development.  If 
s be  supposed  to  var3%  then  / and  z both  must  vary,  and  they  are  all  im- 
plicit functions  of  each  other.  These  coefficients  are  therefore  best 
found  geometrically,  as  follows  : in  Fig.  155, 


Let  AB  — BC  = ds  = differential  portions  of  the  line  AB  = s in  Fig.  154  ; 
AD  — — dt  — change  in  AB  {=  /')  due  to  the  change  + ds  in  s. 

Let  the  angle  BAB  = z'  and  BBC  = z",  z"  being  greater  than  z'  by 
the  convergence  of  the  meridians  shown  by  the  angle  AB' B. 


APPENDIX  D. 


617 


The  lines  BD  and  CE  are  parallels  of  latitude  through  the  points  B and 
C They  cut  all  meridians  at  right  angles. 

Since  the  triangle  ABD  is  a differential  one  on  the  surface  of  the  sphere, 
it  may  be  treated  as  a plane  triangle,  and  we  may  at  once  write 


dt 

ds 


AD 

AB 


— cos  0', 


(10) 


the  minus  sign  indicating  that  I and  s are  inverse  functions  of  each  other. 
Differentiating  this  equation  and  dividing  both  sides  by  ds  we  obtain 


dN  . ,dz'  . . 

(II) 


Now  the  angle  dz’  is  the  angle  AP'B,  subtended  by  the  arc  BD  with 
radius  BP'.  But  this  arc,  with  radius  gives  the  angle  ds  sin  z'. 


Therefore 


dz'  — sin  z ds 


BN 

BP' 


— sin  dds  tan  D = sin  z cot  /Vj, 


dz  . , 

or  — = sin  z cot  / . 

ds  . 

Substituting  this  in  (ii)  we  obtain 

d'^l' 

— sin®^'  cot  /' (12) 

Substituting  these  values  in  (9),  we  have 

I — I = — s cos  2 -f-  sin^  2 cot  / etc. 

Now,  replacing  /,  and  z,  by  L,  L\  and  Z,  we  have 

D — L — s cos  Z -|-  z tan  L (13) 


Here  s is  expressed  in  arc  to  a radius  of  unity. 


6i8 


SI//?  VE  YING. 


Referring  it  now  to  the  radius  N,  we  have  j where  A'  is  the  length 

of  the  arc  s in  any  unit,  iV  being  the  length  of  the  normal  7ivi  in  Fig.  153, 
given  in  the  same  unit. 

Substituting  these  in  (13),  w'e  have 


j,  j.  _ A' cos  Z , I A' 2 sin2  Z tan  L 
” “ ~N  ^2  ' 


(14) 


This  gives  the  difference  of  latitude  in  units  of  arc  in  terms  of  radius  N. 
But  differences  of  latitude  are  measured  on  a sphere  wliose  radius  is 
the  radius  of  curvature  of  the  meridian  at  the  middle  latitude.  Since  we 
do  not  yet  know  the  middle  latitude,  we  can  use  the  known  latitude  L' 

and  afterwards  correct  to  — . 

2 

Changing  to  a sphere  whose  radius  is  R,  and  dividing  by  the  arc  of 
i"  in  order  to  get  the  result  in  seconds,  we  have 


L'  - L = ^ SL  = 


J?  arc  i' 


cos  Z 


If  we  let 


B = 


R arc  1"’ 


and 


C = 


2 I?i\'  arc  I 
tan  A 


- sin®  Z tan  L.  (15) 


2 AW  arc  i"’ 


we  may  write  — oL  = A"  cos  Z-B  AT®  sin®  Z-C. (16) 

To  reduce  this  to  what  it  would  be  if  the  mean  latitude  had  been  used 
we  have  to  correct  it  for  the  difference  in  the  radii  of  curvature.  Rl  and 
Rm,  at  the  latitude  L and  the  middle  latitude  respectively.  If  ZL  be  the 
true  difference  of  latitude  when  Rm  is  used,  and  <5Z  be  the  difference 
when  Rl  is  used,  we  would  have 


ZL  : 8L  ::  Rl  : Rm, 
AL  = = 5Z  (i  ^ 

-A  'itt  ' 


Rr 


)-“(■+ a- 


To  reduce  8L  to  ZL,  therefore,  we  must  add  the  quantity  . 


ciRr 

Rm 


Now 


R 


(I 


a{\  — <?2) 

— ^2  sin®  L)^ 


a{\  — (?2)  {'xe-  sin  A cos  A) 

dR  ~ — ^ dL. 

(I  — e-  sin2  Ajz 


whence 


APPEl^DIX  D. 


6ig 


Here  dL  is  the  difference  in  latitude  between  one  extremity  of  the 
line  s and  its  middle  point,  or  dL  = as  given  in  eq.  (i6),  hence 

(17^ 

\ R1  1—^2  sin^  L 


If  we  put  D — 

the  corrective  term  becomes 


1^2  sin  L cos  L sin 
I — sin*^  L 


= (SL)W; 

Jim 


whence  we  finally  obtain 

- AL  = X cos  Z-B  Z-C+idZy-D,  . , . (D) 

where  SL  is  given  by  (i6),  or  it  is  the  value  of  the  first  two  terms  in  the 
right  member  of  this  final  equation.  For  distances  less  than  I2  miles 
the  first  term  only  may  be  used  as  giving  the  value  of  dL. 

The  values  of  the  constants  B,  C,  and  D are  given  for  every  minute 
of  latitude  from  23°  to  65°  in  Appendix  No.  7 of  the  U.  S.  Coast  and 
Geodetic  Report  for  1884.  This  Appendix  can  be  obtained  by  applying 
to  the  Superintendent. 

For  distances  of  12  miles  or  less,  using  the  first  term  only  for  5Z, 
equation  (18)  becomes 

AL  ^ K cos  Z{B  ^ K cos  Z>D)-Y  sm^  Z-C.  . . . (E') 


THE  DIFFERENCE  OF  LONGITUDE. 

In  the  triangle  APB,  Fig.  154,  the  three  sides  and  the  angle  at  the 
known  station  B are  known.  To  find  AM  = angle  APB,  we  have, 
therefore, 

sin  PA  : sin  ::  sin  PBA  : sin  APB, 
or  sin  / ; sin  s sin  z : sin  AM. 


* In  the  U.  S.  Coast  and  Geodetic  Survey  Report  for  1884,  Appendix  7,  p. 
326,  this  term  is  given  with  its  denominator  raised  to  the  f power,  and  the 
tabular  values  of  D are  computed  accordingly.  The  development  there  given 
is  laborious  and  approximate,  but  the  error  is  not  more  than  0.001  of  the  value 
of  this  term,  which  is  itself  very  small. 


620 


SURVEYING. 


But  j = ^ where  N is  the  normal  An,  Fig.  154;  and  if  we  assume 
that  the  arc  s is  proportional  to  its  sine,  we  have 


AM  = 


K sin  Z 


N cos  L arc  i"’ 
where  AM  is  expressed  in  seconds  of  arc. 

T 

If  we  put  A = 


. (18) 


this  equation  becomes 


Am  = 


N arc  i'" 

A'  sin  Z . A 
cos  L 


fF) 


In  order  to  correct  for  the  assumption  that  the  arc  is  proportional  to 
its  sine,  a table  of  the  differences  of  the  logarithms  of  arcs  and  sines  is 
given  in  the  U.  S.  C.  and  G.  Report  for  1884,  p,  373,  with  instructions 
for  its  use  on  p.  327. 


THE  DIFFERENCE  OF  AZIMUTH. 

In  the  spherical  triangle  APB,  Fig.  154,  we  have,  from  spherical 
trigonometry, 


or 


cot  \{PBA  + PAB)^  = tan  \BPA 


cos  \{BP  AP) 
cos  \{BP  — AP)' 


cot  \{z'  -\~  z)  = tan  ^ (—  AM\) 


cosi(/'-f-/) 
cosi(/'—  /) 


, ^ , sin  i(Z'  -f  Z) 

• = - tan  iAM 7777^— 

cos  i(Z  — Z) 

But  z = 180°  — Z, 

therefore  cot  i(i8o°  — Z z')  = tan  4(Z  — z')  = tan  \{AZ), 

.sin  W ■\-L) 


whence 


tan  ^AZ  = tan  i AM- 


cos  i(Z'  - Z)* 


. (19) 


*Chauvenet’s  Spherical  Trigonometry,  eq.  (127). 
f Increments  of  M are  measured  positively  towards  the  west. 


APPENDIX  D, 


621 


It  will  be  seen  that  since  the  azimuth  Z of  a line  is  measured  from  the 
south  point  in  the  direction  S.W.N.E.,  the  azimuth  of  the  line  BA 
from  B towards  A (forward  azimuth)  is  the  angle  PBA  + 180°  = Z', 
while  the  azimuth  of  the  same  line  from  A is  1^0°  — PAB  = Z.  Also, 
that  ZZ  = Z+  180°  - Z'. 

Assuming  that  the  tangents  ^AZ  and  ^AM  are  proportional  to  their 
arcs,  and  putting  Lm  for  the  middle  latitude,  we  have 


- AZ- 


Am 


sin  Lm 
cos  ^AL 


(G) 


The  U.  S.  Coast  Survey  Tables  are  based  on  the  following  semi- 
diameters: 


a = 6 378  206  metres, 

= 6 356  584  “ 

or  a\b  w 294.98  : 293.98. 

See  Appendix  No.  7,  U.  S.  Coast  and  Geodetic  Survey,  for  tabulaf 
values  of  constants  and  forms  for  reduction. 


TABLES. 


TABLE  I. 

Trigonometric  Formulae. 


Trigonometric  Functions. 


Let  A (Fig.  107)  = angle  BAG  = arc  BF,  and  let  the  radius  AF  = AB  — 
AH=\. 


We  then  have 

sin  ^ = BG 

• cos^  ' = AG 

tan  A — DF 

cot  A = HG 

sec  A = AD 

cosec  A = AO 
versin  A - CF  = BE 
covers  A = BK  ~ HL 
exsec  A = BD 
coexsec  A = BG 
chord  A — BF 
chord  2 A = BI  = 2BC 


Fig.  107. 


In  the  right-angled  triangle  ABC  (Fig.  107) 
Let  AB  = c,  AC  = b,  and  BG  = a. 

We  then  have : 


1. 

sin  A = 

a 

c 

= cosB 

11. 

a 

= c sin  A = b tan  A 

2. 

cos  A — 

b 

c 

= sin  B 

12. 

b 

= c cos^  = a cot  A 

3. 

tan  A = 

a 

F 

= cotB 

13. 

c 

_ a _ & 

~ sin  A “ cos  J. 

4. 

cot  A = 

b 

a 

= tan  B 

14. 

a 

= c cos  B = b cot  B 

5. 

sec  A = 

c 

F 

= cosec  B 

15. 

b 

= c sin  B = a tan  B 

6. 

cosec  A = 

c 

a 

= sec  B 

IG. 

c 

a b 

cos  B ~ sin  B 

7. 

vers  A = 

c — b 

c 

— covers  B 

17. 

a 

= V(c+6)(c-0) 

8. 

exsec  A = 

c - b 
b 

= coexsec  B 

18. 

b 

= 4^  (c  -f  a)  (c  - a) 

9. 

covers  A = 

c — a 

c 

= versin  B 

19. 

c 

= Vaa-iT^i 

10. 

coexsec  A = 

c — a 

a 

= exsec  B 

20. 

C 

= 00»  = ^ + 2? 

21.  area 

ah 

626 


SURVE  YING. 


TA'BLE  I. — Continued. 

Tfuoonometric  Formul.^. 


Solution  oir  Oblk^uk  Trianoles. 


GIVEN. 

SOUGHT. 

FORMUL.1!:.  1 

1 

22 

A.  D,  a 

C,  b,  c 

c = 180°  - (y1  -1-  77),  b = - . sin  77. 

sin  A 

j 

23 

A,  a,  6 

1 

r:;i  n = J,,  C=  ISO"  - (A  + 1!). 

j 

c = sin  C. 

sin  A 

24 

C,  a,  h 

1 4-  B) 

y.  (^  + 77)  = C0°  - 14  C 

25 

tan  ^ M - 77)  = tan  14  M + 77) 

26 

A,  B 

A = 14  {A  + 77)  -f  14  M - B), 

B^y  {A  + B)-y  (A  - B) 

27 

c 

‘ Qosy{A—B)  ^ sin  y(A  — B) 

28 

area 

77  = 34  rt  & sin  C. 

29 

a,  by  c 

A 

Let  s = 34  (a  6 -f  c) ; sin  34 % c"  ”” 

30 

1/  .1  /s(s  — a)  . . /(s-b)(s-c) 

cos  ^ ; tan  .1  ^ 

81 

sin  A ~ C-5— — c)_ 

~ be  ' 

2 (.9  -b)  (s-  c) 

vers  ^ = , 

be 

32  ® 

area 

K = Vs  (s  — a)  (s  — 6)  (.;  — c) 

33 

Ay  B,  C,  a 

area 

sin  77 . sin  C 
~ 2 sin 

TABLES. 


627 


TABLE  I. — Contmued. 

TrigonOxMetric  Formula. 

general  formula. 


34  sin 


= VI  — cos2  A = tan  A cos  A 


36 

37 

38 

39 

40 

41 

42 

43 

41 

45 

40 

47 

48 

49 

50 

61 


sin  A 
sin  A 

cos  A 

cos^ 
cos  A 
tan  A — 

tan  A = 

tan  A = 

cot  A ~ 

cot  A = 

cot  A =s 


cosec  A 

2 sin  14  A cos  A = vers  A cot  14 


Y 54  vers  2 A = (1  — cos  2 A) 

1 


= V 1 — sin2  A = cot  A sin  A 


sec  A 

1 — vers  A = 2 cos^  = 1 — 2 sin^  14  ^ 


cos^  y^A  — sin2  yA  = V cos  2 A 

1 sin  A 


cot  A cos  A 


= V seo^  A — I 


— 1 


cos^  A 
1 — cos  2 A vers  2 A 


V ^ — cos‘^  ^4  _ sin  2 A 
cos  A i cos  2 A 


sin  2 A 


sin  2 A 


= exsec  A cot  J4  A. 


cos  A 


— — j = — — / = i/cosec^J.  — 1 

tan  A sin  A 


sin  2 A 
1 — cos  2 A 

_tan  J4  A 
exsec  A 


sin  2 A 
vers  2 A 


1 + cos  2 A 
sin  2 A 


vers  A = 1 — cos  A = sin  A tan  yA  = 2 sin^  y A 

vers  A = exsec  A cos  A 

exsec  A = sec  A — 1 = tan  A tan  yA  = 

cos  A 


sin  yA 
sin  2 A = 2 sin  A cos  A 


/I  — cos  A / versyl 

~ y 2 ~ y 2 


cos  Yi  A 


s/'- 


+ cos^ 
2 


cos  2 A = 2 cos^  A — 1 = cos^  A — sin^  yl  = 1 — 2 sin^'  A 


628 


SURVEYING. 


53. 

54. 

65. 

56. 

57. 

68. 

69. 

60. 

61. 

62. 

63. 

64. 

65. 

66. 

67. 

68. 

69. 

70. 


TABLE  I. — Continued. 
Trigonometric  Formula. 


General  FoRMCLiE. 


. . , . tan  A 

tan  7—r  ^ = cosec 

1 4-  sec  A 


A ^ A 1 ^ A / ^ 

A — cot  A = . . — - = -4/  j- 

sm  ^ r I 


— cos 
4- cos 


tan  2 A = 


cot.  = 


cot  2 ^ = 


vers  ^ A ■ 


2 tan  A 
1 — tan^  A 

_Bln  A__  _ 1 4-  cos  A 
vers  k “ sin  A 

^ot2  A — 1 
2 cot  .4 

^ vers  A 


cosec  A — cot  A 


1 — cos  A 


1 4-  — 34  vers  A 24-  ^2  (1  4-  cos  A) 

vers  2^  = 2 sin^  A 


exsec  }^A=- 


— cos  A 


exsec  2 A = 


(1  4-  cos  A)  + 4/2  (1  4-  cos  A) 
tan^  A 


1 — tana  ^ 
sin  ± S)  = sin  A . cos  5 ± sin  .B , cos  A 
cos  {A  ± B)  = cos  A . cos  B T sin  ^ . sin  5 
sin  ^ 4-  sin  j5  = 2 sin  (4.  4-  B)  cos  34  (4  — B) 
sin  4 — sin  B = 2 cos  34  (4  4-  B)  sin  34  (4  — B) 
cos  4 4-  cos  B = 2 cos  y^{A-\-  B)  cos  34  (4  — B) 
cos  B — cos  4 = 2 sin  34  (-4  4-  -S)  sin  ^(A  — B) 
sina  4 — sina  B = cosa  B — cosa  4 = sin  (4  4-  -B)  sin  (4  — 
cosa  4 — sina  ^ = ^os  (4  4*  i?)  cos  (4  — B) 
sin  (A  4-  B) 


tan  4 4-  tan  B = 


tan  4 — tan  B 


cos  4 . cos  B 

sin  (4 B) 
cos  4 . cos  B 


TABLES, 


629 


TABLE  11. 

For  Converting  Metres,  Feet,  and  Chains. 


Metres  to  Feet. 

Feet 

TO  Metres  and 

Chains. 

Chains 

TO  Feet. 

Metres. 

Feet. 

Feet. 

Metres. 

Chains. 

Chains. 

Feet. 

I 

3.28087 

I 

0.304797 

O.OI51 

0.01 

0.66 

2 

6.56174 

2 

0.609595 

-0303 

.02 

1.32 

3 

9.84261 

3 

0.914392 

-0455 

-03 

1.98 

4 

13.12348 

4 

1.219189 

.0606 

.04 

2.64 

5 

16.40435 

5 

1.523986 

-0758 

-05 

3-30 

6 

19.6S522 

6 

I .828784 

.0909 

.06 

3.96 

7 

22.96609 

7 

2.133581 

. 1061 

-07 

4.62 

8 

26.24695 

8 

2.438378 

.1212 

.oS 

5.28 

9 

29-52732 

9 

2.743175 

.1364 

.09 

5.94 

10 

32 . 80869 

10 

3.047973 

-1515 

. 10 

6.60 

20 

65.61739 

20 

6.095946 

.3030 

.20 

13.20 

30 

98.42609 

30 

9.143918 

-4545 

-30 

19.80 

40 

131.2348 

40 

12. 19189 

.6061 

.40 

26.40 

50 

164.0435 

50 

15.23986 

.7576 

-50 

33.00 

60 

196.8522 

60 

18.28784 

.9091 

.60 

39.60 

70 

229.6609 

70 

21.33581 

1.0606 

-70 

46.20 

80 

262.4695 

80 

24-38378 

1 .2121 

.80 

52.80 

90 

295.2782 

90 

27-43175 

1.3636 

.90 

59-40 

100 

328.0869 

100 

30.47973 

I. 5151 

I 

66.00 

200 

656.1739 

100 

60.95946 

3-0303 

2 

132 

300 

984.2609 

300 

91.43918 

4-5455 

3 

198 

400 

1312.348 

400 

121.9189 

6 . 0606 

4 

264 

500 

1640.435 

500 

152.3986 

7.5756 

5 

330 

600 

1968.522 

600 

182.8784 

9 . 0909 

6 

396 

700 

2296.609 

700 

213-3581 

10.606 

7 

462 

8 00 

2624.695 

800 

243.8378 

12. 121 

8 

528 

900 

2952.782 

900 

274-3175 

13.636 

9 

594 

1000 

3280.869 

1000 

304 • 7973 

15-151 

10 

660 

2000 

6561.739 

2000 

609.5946 

30.303 

20 

1320 

3000 

9842.609 

3000 

914.3918 

45-455 

30 

1980 

4000 

13123.48 

4000 

1219. 189 

60 . 606 

40 

2640 

5000 

16404.35 

5000 

1523.986 

75.756 

50 

3300 

6000 

19685.22 

6000 

1828.784 

90.909 

60 

3960 

7000 

22966.09 

7000 

2133-581 

106.06 

70 

4620 

8000 

26246.95 

8000 

2438.378 

121 .21 

80 

5280 

9000 

29527.82  , 

9000 

2743-175 

136.36 

90 

5940 

630 


S UR  VE  YJNG. 


TABLE  III. 

Logarithms  of  Numbers.  § 173. 


c/i 

0 

rt 

0 

1 

a 

3 

4 

6 

G 

7 

8 

9 

Proportional  Parts. 

>' 

2 

1 

.1 

\ 

i 

1 

G 

1 1 

7 

H 

0 

10 

.0000 

.0043 

.0086 

.0128 

.0170 

.0212 

•0253 

.0294 

•0334 

i 

•0374 

4 

Is 

t2 

. 

: 

i 

25  29 

33 

37 

II 

.0414 

.0453 

.0492 

•0531 

.056a 

.0607 

.0645 

.0682 

.0719 

•0755 

4 

8 

I I 

i5>9 

23  26 

30 

34 

12 

.0792 

.0828 

.0864 

.0899 

•0934 

.0969 

. 1CX34 

. 1038 

. 1072 

. 1 106 

3 

1 7 

10 

14  >7 

21 

24 

28 

3> 

.”39 

.1173 

.1206 

.1239 

.1271 

•1303 

• 1335 

.1367 

.1399 

.1430 

3 

6 

10 

•3, 

iq  23 

26 

M 

. 1461 

.1492 

• 1523 

• 1553 

.1584 

. 1614 

.1644 

.1673 

.1703 

•»732 

3 

6 

9 

12 

15 

18 

21 

24 

'27 

1 

15 

.1761 

.1790 

.1818 

.1847 

• 1875 

.1903 

• 1931 

.1959 

.1987 

.2014 

3 

6 

1 8 

1 1 

14 

17  20 

22 

125 

16 

.2041 

.2068 

• 2095 

.2122 

.2148 

•2175 

.2201 

.2227 

.2253 

.2279 

3 

5 

8 

1 1 

»3 

16  18 

21 

'24 

>7 

.2304 

.2330 

•2355 

• 2380 

.2405 

.243d 

•2455 

.2480 

.2504 

.2529 

2 

5 

1 7 

10 

15  »7  20 

22 

18 

.2553 

•2577 

.2601 

.2625 

.2648 

.267s 

•2695 

.2718 

•2742 

.2765 

2 

5 

1 7 

1" 

'4 

16 

,»9 

21 

19 

.2788 

.2810 

.2833 

.2856 

.2878 

.2900 

.2923 

•2945 

.2967 

.2989 

2 

4 

1 ^ 

9 

1 1 

'3 

16  18 

20 

i 

20 

.3010 

.3032 

• 3054 

• 3075 

.3096 

.311S 

•3139 

.3160 

• 318. 

• 3201 

2 

4 

l6 

8 

1 1 

'3 

'.5 

17  19 

21 

.3222 

•3243 

•3263 

• 3284 

•3304 

•3324 

•3345 

•3365 

•3385 

• 3404 

2 

4 

‘ 6 

8 

10 

12 

M 

'16  j8 

22 

•3424 

•3444 

.3464 

• 3483 

.3502 

•3522 

•354> 

•3560 

•3579 

•3598 

1 2 

4 

1 6 

8 

10 

12  14  15 

'7 

23 

•3617 

.3636 

.3655 

•3674 

.3692 

•37” 

•3729 

•3747 

.3766 

•3784 

i 2 

4 

6 

7 

9 

1 1 

l'3 

!‘5 

'7 

24 

.3802 

.3820 

•3838 

.3856 

•3874 

.3892 

•3909 

•3927 

•3945 

.3962 

1 ^ 

4 

5 

7 

9 

[ 1 

16 

i 

25 

•3979 

•3997 

.4014 

.4037 

.4048 

• 4065 

.4082 

.4099 

.4116 

•4133 

i ^ 

3 

5 

7 

9 

10 

1 

12 

1 

'5 

26 

.4150 

.4166 

.4183 

.4200 

.4216 

.423. 

•4249 

.4265 

.4281 

.4298 

2 

3 

5 

7 

8 

10  II 

13 

15 

27' 

•4314 

•4330 

.4346 

.4362 

•4378, 

•4393 

.4409 

•4425 

.4440 

.4456 

1 

3 

5 

6 

8 

9 II 

13 

M 

28 

.4472 

•4487 

.4502 

.4518 

•4533 

•4548 

.4564 

•4579 

•4594 

.46^ 

3 

5 

6 

8 

9 

11 

12 

l«4 

29 

.4624 

•4639 

.4654 

.4669 

.46^3 

.4698 

•4713 

.4728 

.4742 

•4757 

j ' 

3 

4 

6 

7 

9 10 

12 

1 

,13 

30 

•4771 

.4786 

.4800 

.4814 

.4829 

•4843 

•4857 

.4871 

.4886 

.4900 

3 

4 

6 

7 

9 10 

1 

II 

13 

3^ 

.4914 

.4928 

.4942 

• 4955 

.4969 

•4983 

•4997 

.5011 

.5024 

.5038 

3 

4 

6 

7 

8 

10 

11 

,12 

32, 

.5051 

.5065 

•5079 

.5092 

•5105 

•5”9 

•5132 

•5145 

•5159 

•5172 

; I 

1 3 

4 

5 

7 

8 

9 

II 

12 

33 

.5185 

.5198 

.5211 

• 5224 

•5237 

.5250 

.5263 

.5276 

.5289 

.5302 

I 

3 

i ^ 

5 

6 

8 

9 

'10 

12 

34 

•5315 

•5328 

•5340 

•5353 

.5366 

•5378 

•5391 

•5403 

.5416 

•5428 

I 

1 

1 ^ 

4 

5 

6 

8 

9 

II 

35 

•5441 

•5453 

•5465 

• 5478 

•5490 

•5502 

•5514 

•5527 

•5539 

•5551 

I 

2 

4 

5 

6 

7 

9 10 

36, 

•5563 

•5575 

•5587 

•5599 

.5611 

.5623 

•5635 

•5647 

.5658 

.5670 

I 

2 

! 4 

5 

6 

7 

8 

10  II 

37 

.5682 

•5694 

• 5705 

• 5717 

•5729 

•5740 

•5752 

•5763 

•5775 

•5786 

I 

2 

3 

5 

6 

7 

8 

9 

10 

38, 

•5798 

.5809 

.5821 

• 5832 

•5843 

•5855 

.5866 

•5877 

.5888 

•5899 

I 

2 

3 

5 

6 

7 

i 8 

9 

10 

39 

•59” 

.5922 

•5933 

• 5944 

•5955 

.5966 

•5977 

.5988 

•5999 

.6010 

I 

2 

3 

4 

5 

7 

8 

9 

10 

40 

.6021 

.6031 

.6042 

• 6053 

.6064 

.6075 

. 6085 

.6096 

.6107 

.6117 

I 

2 

I 3 

4 

5 

6 

8 

9 

10 

41 

.6128 

.6138 

.6149 

.6160 

.6170 

.6180 

.6191 

.6201 

.6212 

.6222 

1 

2 

3 

4 

5 

6 

7 

8 

9 

42 

.6232 

.6243 

.6253 

.6263 

.6274 

.6284 

.6294 

.6304 

•6314 

.6325 

I 

2 

3 

4 

5 

6 

7 

8 

9 

43 

•6335 

•6345 

.6355 

•6365: 

•6375 

.6385 

•6395 

.6405 

.6415 

.6425 

I 

2 

3 

4 

5 

6 

7 

8 

9 

44 

•6435 

.6444 

• 6454 

.6464 

.6474 

.6484 

•6493 

.6503 

•6513 

• 6522 

2 

3 

4 

5 

6 

7 

8 

9 

45 

•6532 

.6542 

• 6551 

• 6561 

•657’ 

.6580 

.6590 

•6599 

.6609 

.6618 

I 

2 

3 

4 

5 

6 

7 

8 

9 

46 

.6628 

.6637 

.6646 

.6656 

.6665 

.6675 

.6684 

.6693 

.6702 

.6712 

I 

2 

3 

4 

5 

6 

7 

7 

8 

47 

.6721 

.6730 

• 6739 

.6749 

.6758 

.6767 

.6776 

.6785 

.6794 

.6803 

I 

2 

3 

4 

5 

5 

6 

7 

8 

48, 

.6812 

.6821 

.6830 

.6839 

.6848 

.6857 

.6866 

.6875 

.6884 

.6893 

I 

2 

3 

4 

4 

5 

6 

7 

8 

49 

.6902 

.6911 

.6920 

.6928 

•6937 

.6946 

•6955 

.6964 

.6972 

.6981 

I 

2 

3 

4 

4 

5 

6 

7 

8 

50 

.6990 

.6998 

.7007 

.7016 

.7024 

•7033 

.7042 

.7050 

•7059 

.7067 

I 

2 

3 

3 

4 

5 

6 

7 

8 

51 

.7076 

.7084 

• 7093 

.7101 

.7110 

.7118 

.7126 

•7135 

•7143 

•7152 

I 

2 

3 

3 

4 

5 

6 

7 

8 

52 

.7160 

.7168 

.7177 

• 7183 

•7193 

.7202 

.7210 

.7218 

.7226 

•7235 

I 

2 

2 

3 

4 

5 

6 

7 

7 

53 

•7243 

.7251 

• 7259 

.7267 

•7275 

•7284 

.7292 

.7300 

.7308 

•7316 

I 

2 

2 

3 

4 

5 

6 

6 

7 

54' 

•7324 

•7332 

• 7340 

.7348 

.735f 

•7364 

•7372 

.7380 

.7388 

•7396 

I 

2 

2 

3 

4 

5 

6 

6 

7 

TABLES. 


631 


TABLE  III. — Continued. 

Logarithms  of  Numbers. 


Nat.  Nos,  1 

0 

1 

3 

3 

4 

5 

6 

7 

8 

9 

Proportional  Parts. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

,S5 

.7404 

•74*2 

•74*9 

• 7427 

•7435 

•7443 

•745* 

•7459 

.7466 

•7474 

1 

2 

2 

3 

4 

5 

5 

6 

7 

5t> 

.7482 

.7490 

•7497 

•7505 

•75*3 

•7520 

• 7528 

•7536 

•7543 

•755* 

I 

2 

2 

3 

4 

5 

5 

6 

7 

57 

•7559 

.7566 

•7574 

• 7582 

•7589 

•7597 

.7604 

.7612 

.7619 

•7627 

I 

2 

2 

3, 

4 

5 

5 

6 

7 

58 

•7634 

.7642 

.7649 

•7657 

.7664 

.7672 

.7679 

.7686 

.7694 

.770*1 

I 

I 

2 

3 

4 

4 

5 

6 

7 

59 

.7709 

•77*6 

•7723 

•773* 

•7738 

•7745 

•7752 

.7760 

•7767 

•7774: 

1 

2 

3 

4 

4 

5 

6 

7 

60 

.7782 

•7789 

.7796 

•7803 

.7810 

.7818 

•7825 

•7832 

•7839 

.7846 

I 

I 

2 

3 

4 

4 

5 

6 

6 

61 

•7853 

.786c 

.7868 

•7875 

.7882 

.7889 

.7896 

•7903 

•79*0 

•79*7 

I 

I 

2 

3 

4 

4 

5 

6 

6 

62 

.7924 

•793* 

•7938 

•7945 

•7952 

•7959 

.7966 

■7973 

.7980 

.7987 

I 

I 

2 

3j 

3 

4 

5 

6 

6 

63 

•7993 

.8000 

.8007 

.8014 

.8021 

.8028 

•8oj5 

.8041 

.8048 

•8055 

I 

I 

2 

3 

3 

4 

5 

5 

6 

64 

.8062 

.8069 

•8075 

.8082 

.8089 

.8096 

.8102 

.8109 

.8116 

.8122 

I 

2 

3 

3 

4 

5 

5 

6 

65 

.8129 

.8136 

.8142 

.8149 

.8156 

.8162 

.8169 

.8176 

.8182 

.8189 

I 

I 

2 

3 

3 

4 

5 

5 

6 

66 

•8195 

.8202 

.8209 

.8215 

.8222' 

.8228 

•8235 

. .8241 

.8248 

•8254 

I 

I 

2 

3 

3 

4 

5 

5 

6 

67 

.8261 

.8267 

.8274 

.8280 

.8287' 

.8293 

.8299 

• 8306 

• 8312 

•8319 

I 

I 

2 

3 

3 

4 

5 

5 

6 

68 

•8325 

■8331 

•8338 

1 -8344 

•835* 

•8357 

•8363 

•8370 

.8376 

■83821 

I 

I 

2 

3 

3 

4 

4 

5 

6 

69 

.8388 

•8395 

.8401 

.8407 

.8414 

0 

00 

.8426 

•8432 

•8439 

•8445 

I 

2 

2 

3 

4 

4 

5 

6 

70 

.8451 

•8457 

.8463 

.8470 

.8476 

. 8482 

.8488 

•8494 

. 8500 

■ 8506 

I 

I 

2 

2 

3 

4 

4 

5 

6 

71 

•85'3 

•8519 

•8525 

•853' 

•8537, 

• 8543 

•8549 

•8555 

• 8561 

•85671 

I 

I 

2 

2 

3 

4 

4 

5 

5 

72 

•8573 

■•8579 

•8585 

•859* 

•8597, 

.8603 

.8609 

.8615 

.8621 

.8627' 

I 

I 

2 

2 

3 

4 

4 

5 

5 

73 

.8633 

.8639 

.8645 

.8651 

.8657 

.8663 

.8669 

.8675 

.8681 

.8686, 

I 

I 

2 

2 

3 

4 

4 

5 

5 

74 

.8692 

.8698 

.8704 

.8710 

.8716 

.8722 

.8727 

•8733 

•8739 

•8745'; 

I 

2 

2 

3 

4 

4 

5, 

5 

75 

•8751 

.8736 

.8762 

.8768 

•8774 

• 8779 

•8785 

■879* 

•8797 

.8802* 

T 

r 

2 

2 

3 

3 

4 

5 

5 

76 

.8808 

.8814 

.8820 

1 .8825 

•8831 

•8837 

.8842 

.8848 

• 8854 

•8859^ 

I 

I 

2 

2 

3 

3 

4 

5 

5 

77 

.8865 

.8871 

.8876 

.8882 

.8S87 

•8893 

.8899 

.8904 

.8910 

•8915' 

I 

I 

2 

2 

3 

3 

4 

4 

5 

78 

.8921 

•8927 

.8932 

• 8938 

•8943 

• 8949 

•8954 

.8960 

.8965 

.89711 

I 

I 

2 

2 

3 

3 

4 

4 

5 

79 

.8976 

.8982 

.8987 

•8993 

.8998 

.9004 

.9009 

.9015 

.5020 

•9025 

I 

2 

2 

3 

3 

4 

4 

5 

80 

.9031 

.9036 

.9042 

.9047 

•9053 

.9058 

.9063 

.9069 

.9074 

•9079] 

I 

I 

2 

2 

3 

3 

4 

4 

5 

81 

.9085 

.9090 

.9096 

.9101 

.9106 

.9112 

•9**7 

.9122 

.9128 

■91.33! 

I 

I 

2 

2 

3 

3 

4 

4 

5 

82 

.9138 

•9*43 

•9*49 

•9*54 

•9159' 

.9165 

.9170 

•9*75 

.9180 

.9186^ 

I 

1 

2 

3 

3 

4 

4 

5 

83 

.9191 

.9196 

.9201 

.9206 

.9212 

• 9217 

.9222 

•9227 

•9232 

•9238I 

I 

■ I 

2 

2 

3 

3 

4 

4 

5 

84 

•9243 

.9248 

•9253 

I -9258 

.9263 

.9269 

.9274 

•9279 

.9284 

•9289 

^1 

I 

2 

2 

3 

3 

4 

4 

5 

85 

.9294 

•9299 

•9304 

.9309 

•93*5 

.9320 

•9325 

■9330 

•9335 

1 

•9340^ 

1 

1 

1 I 

2 

2 

3 

3 

4 

4 

5 

86 

•9345 

•9350 

•9355 

.9360 

•9365, 

•9370 

■9375 

•9380 

•9385 

■9390 

^1 

I 

2 

2 

3 

3 

4 

4 

5 

87 

•9395 

.9400 

.9405 

•94*0 

•94*5‘ 

.9420 

•9425 

•9430 

•9435 

■9440' 

0; 

I 

I 

2 

2 

3 

3 

4 

4 

83 

•9445 

•9450 

•9455 

1 . 9460 

•9465 

.9469 

•9474 

■9479 

.9484 

•9489' 

I 

I 

2 

2 

3 

3 

4 

4 

89 

•9494 

•9499 

.9504 

! -9509 

•95*3 

•95*8 

•9523 

•9528 

•9533 

•9538, 

0! 

1 

2 

2 

3 

3 

4 

4 

90 

•9542 

•9547 

•9552 

•9557 

.9562 

.9566 

•957* 

.9576 

•9581 

•9586 

1 

0 

I 

I 

2 

2 

3 

3 

4 

91 

•9590 

•9595 

.9600 

.9605 

. 9609 

.9614 

.9619 

.9624 

.9628 

■96.33; 

0 

I 

I 

2 

2 

3 

3 

4 

4 

92 

.9638 

•9643 

.9647 

9652 

•9657 

.9661 

.9666 

.9671 

•9675 

.9680 

0 

I 

I 

2 

2 

3 

3 

4 

4 

93 

.9685 

.9689 

.9694 

.9699 

•9703 

.9708 

•97*3 

•97*7 

•9722 

•9727 

0 

I 

1 

2 

2 

3 

3 

4 

4 

94 

•973* 

•9736 

.9741 

•9745 

•9750 

•9754 

•9759 

•9763 

.9768 

•9773 

0 

I 

2 

2 

3 

3 

4 

4 

95 

•9777 

.9782 

.9786 

.9791 

•9795 

.9800 

.9805 

.9809 

.9814 

.9818 

0 

I 

I 

2 

2 

3 

3 

4 

4 

96 

.9823 

.9827 

•9832 

.9836 

.9841 

9845 

.9850 

■9854 

•9859 

.9863 

0 

I 

I 

2 

2 

3 

3 

4 

4 

97 

.9868 

.9872 

•9877 

.9881 

.9886 

.9890 

•9894 

•9899 

•9903 

.9908 

0 

I 

I 

2I 

2 

3 

3 

4 

4 

98 

•99*2 

•99*7 

.9921 

.9926 

•9930 

•9934 

•9939 

•9943 

.9948 

•9952 

0 

I 

I 

2 

2 

3 

3 

4 

4 

99 

•995^ 

.9961 

•9965 

.9969 

•9974 

•9978 

•9983 

•9987 

•999* 

•9996 

I 

2 

2 

3 

3 

3 

4 

Logarithmic  Traverse  Table.  § 173. 

Zero  angle  at  South  Point,  and  increasing  to  W.  (90“),  N.  (180®),  E.  (270®). 


632 


SU/>!  VE  YING. 


TABLES. 


633 


A 000  m m n M A oco  r^vo 

W MMMW 

I 


0^00  10  • 


o 

^0  VC  VO  VO 
0 0 0-0^ 
O O'  C>  3'  O 
On  OV  O'  Q 

c5 


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O'  O'  O 0"0  O'  O'  Os  o>  ^ ^ w-  w w ^ 

O'O  O'O'O'O'OO'O'  v3  O'O'O'O'O'O'O'OsO'  v3 
O'OsOsO'O'O'OsO'O'  ^ O'O'OsO'OsO'OsO'O'  ^ 


O 

0'0s0'c>0'0s0s0'0s  O 

O'.  OsO'O'OsOsO'OsO'  Q 


O'  O'  O'CO  rts 
r--  O roso 
CN  fO  ro  ro  Cv 
VC  VO  'O  VO 


O 

ro  O'  .1** 
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10  10  •O'O 'O  w t^oo  00  00  00  ^ O 

VO'OvOvO'O'OvO'OVO  ^ 'OvOVOvO'O'OsO'OvO  ^ 


00 

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vovo 


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€v  w C4  (N  - 


VX^  ^ 0 


CO  CO  fo  fo  CO  cn  fo  rc  CO 


lovo  t>.oo  O'  0 


O'CO  t^vo  10  • 


O'CO  t^vo  10  fO  N H 0 O'©©  tN.VO  10  CO  N 

: ^ . i 


o 

O'  O'  O'  O'  ^ 
O'  O'  O'  O'  i: 
O'  O'  O'  O'  w5 
O'  O'  O'  OS  Q 


CO  00  00  00  00  00  00  00  O©  Ci 
O'OO'O'OS'O'O'O'O'  ^ 
O'O'O'OsO'O'O'OsO'  ^ 
OsO'O'O'O'OsOsO'Os  ^ 


00  00  ?0  00  00  00  00  00  CO  ffi 
O'O'Osa'OsOsO'O'Os  j: 
O'O'O'O'O'O'O'O'O'  vS 
OO'O'OvO'O'O'OvO.  gi 


O'O'OsD'O'^'O'O'O'  iT 
O'O'O'O'C'O'OsO'O'  w5 

<>c>(>c^0'0><>0>0>  01 


C^  t>.00  00  00  O'  O'  o O 


O O'  t^vo  « CO 
O'  M VC  O '^00  •- 


loioiotnioioioioj 


VO  t'^oo  O'  0 
« N C M 


fo  CO  CO  fo  fo  CO 


\o  c^oo  o^  o w 


.2  © 


000  10  Tj-  CO  N M Q 

<NNCINC<(NCIC<C<  ^ 

I 


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0.00  t^.O  lo  ■.!•  CO  I 

I 


O'  O'  O'  O'  O'  O'  O'  C) 


O.COIO.O'O'O.  01 

O' fli 


CO 

m o'.©  2 

. O « PI  "P 

00  O'  O'  Cl 


w O'  N PI  00  O 
lOOO  PI  10  t'-  o 
lO'O  00  O'  Q cc 
O'  O'  O'  O'  0 o 


>0  P~  t".  tv.  VO  .o 


O'  »ooo  O O' 


0000 

q6 


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tN.  0 M M Q\VO  H SO 

Cl 

CO 

f:. 

O'CO  VO  H O'  C^ 

t>»00  Os  0 M *-i  (s  cc 

H 

M 

ww>-«WNC<C»CV 

N 

06 

00 

VO  tvoo  O'  o 

W N M Pt  „ 


O'  o 

»0 


TABLE  \\ .—Continued. 

Logarithmic  Traverse  Table. 

Zero  angle  at  South  Point,  and  increasing  to  W.  (90°),  N.  (i8o°\  E.  (270°). 


634 


SUR  VE  YIi\G. 


TABLES. 


635 


Logarithmic  Traverse  Table. 

Zero  angfle  at  South  Point,  and  increasinf?  to  W.  (90°),  N.  (180°),  E.  (270®). 


636 


•Vt/A'  VE  YIKC. 


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SURVEY  INC. 


TABI.E  V. 


Horizontal  Distances  and  Elevations  from  Stadia  Readings.  § 204, 


I 

Minutes. 

00 

1 

0 

2° 

30 

Ilor. 

Dinr. 

llor. 

Dinr. 

Hor. 

Dinr. 

Hor. 

Dinr. 

Dist. 

Elcv. 

Dist. 

Elcv. 

Dist. 

Idcv. 

Dist. 

Elcv 

0 . . 

100.00 

0.00 

99-97 

1.74 

99.88 

3-49 

99-73 

5-23 

2 . . 

“ 

0.06 

“ 

1.80 

99.87 

3-55 

99.72 

5.28 

4 • • 

a 

0.12 

ii 

1.86 

“ 

3.60 

99.71 

5-34 

6 . . 

u 

0.17 

99.96 

1.92 

<( 

3-66 

“ 

S-40 

8 . . 

4( 

0.23 

“ 

1.98 

99.S6 

3-72 

99.70 

5-46 

10  . . 

<c 

0.29 

“ 

2.04 

U 

00 

ro 

99-69 

5.52 

12  . . 

“ 

0-35 

2.09 

99-85 

3-84 

5-57 

14  . . 

0.41 

99-95 

2.15 

4< 

3-90 

99.68 

S-63 

16  . . 

<( 

0.47 

“ 

2.21 

99.84 

3-95 

“ 

5-69 

18  . . 

“ 

0.52 

2.27 

“ 

4.01 

99-67 

5-75 

20  . . 

0.58 

“ 

2-33 

99-83 

4.07 

99.66 

5.80 

22  . . 

“ 

0.64 

99-94 

2.38 

4-13 

“ 

5.86 

24  . . 

0.70 

<( 

2.44 

99.82 

4.18 

99-65 

5-92 

26  . . 

99.99 

0.76 

“ 

2.50 

“ 

4.24 

99.64 

5-98 

28  . . 

“ 

o.Si 

99-93 

2.56 

99.81 

4-30 

99-63 

6.04 

30  . . 

“ 

0.87 

“ 

2.62 

“ 

4-36 

“ 

6.09 

32  . . 

0-93 

« 

2.67 

99.80 

4.42 

99.62 

6.15 

34  • • 

0.99 

“ 

2.73 

“ 

4.48 

6.21 

36  . . 

1.05 

99.92 

2.79 

99-79 

4-53 

99.61 

6.27 

38  . . 

4( 

i.i  I 

2.85 

“ 

4-59 

99.60 

6-33 

40  . . 

1. 16 

“ 

2.91 

99.78 

4-65 

99-59 

6.38 

42  . . 

1.22 

99.91 

2.97 

4.71 

U 

6.44 

44  • • 

99.9S 

1.28 

<( 

3.02 

99-77 

4-76 

99-58 

6.50 

46  . . 

(( 

1-34 

99.90 

3.08 

“ 

4.82 

99-57 

6.56 

48  . . 

1.40 

“ 

3-14 

99.76 

4.88 

99-56 

6.61 

50  . . 

1.45 

(( 

3-20 

(( 

4-94 

n 

6.67 

52  . . 

4( 

I-5I 

99.89 

3-26 

99-75 

4.99 

99-55 

6.73 

54  • . 

^•57 

“ 

3-31 

99-74 

5-05 

99-54 

6.78 

56  . . 

99-97 

1.63 

“ 

3-37 

“ 

5-II 

99-53 

6.84 

58  . . 

1.69 

99.S8 

3-43 

99-73 

5-17 

99.52 

6.90 

60  . . 

1.74 

3-49 

5-23 

99-51 

6.96 

^ = 0.75 

0.75 

0.0 1 

0.75 

0.02 

0.75 

0.03 

0.75 

0.05 

r = 1 .00 

1. 00 

0.01 

1. 00 

0.03 

1. 00 

0.04 

1. 00 

0.06 

c — 1.25 

I 25 

0.02 

1.25 

0.03 

1.25 

0.05 

1.25 

1 

0.08 

♦ This  tabic  was  computed  by  Mr.  Artliur  Winslow  of  the  State  Geological  Survey  of  Pennsylvania. 


TABLES. 


641 


TABLE  V. — Continued. 


Horizontal  Distances  and  Elevations  from  Stadia  Readings. 


Minutes. 

4 

.0 

50 

6° 

70 

Hor. 

DifT. 

Hor. 

Diff. 

Hor. 

Difif. 

Her. 

Diff. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

0 . . 

99-51 

6.96 

99-24 

8.68 

98.91 

10.40 

98.51 

12.10 

2 . . 

“ 

7.02 

99-23 

8.74 

98.90 

10.45 

98.50 

12.15 

4 • • 

99.50 

7.07 

99.22 

8.80 

98.88 

10.51 

98.48 

12.21 

6 . . 

99.49 

7-13 

99.21 

8.85 

98.87 

10.57 

98.47 

12.26 

8 . . 

99.48 

7.19 

99.20 

8.91 

98.86 

10.62 

98.46 

12.32 

10  . . 

99-47 

7-25 

99.19 

8.97 

98.85 

10.68 

98.44 

12.38 

12  . . 

99.46 

7-30 

99.18 

9-03 

98.83 

10.74 

98.43 

12.43 

14  . . 

7-36 

99.17 

9.08 

98.82 

10.79 

98.41 

12.49 

16  . . 

99-45 

7-42 

99.16 

9.14 

98.81 

10.85 

98.40 

12.55 

18  . . 

99.44 

7-48 

99-15 

9.20 

98.80 

10.91 

98-39 

12.60 

20  . . 

99-43 

7-53 

99.14 

9-25 

98.78 

10.96 

98.37 

12.66 

22  . . 

99.42 

7.59 

99-13 

9-31 

98.77 

1 1.02 

98.36 

12.72 

24  . . 

99.41 

7-65 

99.11 

9-37 

98.76 

11.08 

98.34 

12.77 

26  . . 

99.40 

7.71 

99.10 

9-43 

98.74 

II. 13 

98.33 

12.83 

28  . . 

99-39 

7.76 

99.09 

9-48 

98-73 

1 1. 19 

98.31 

12.88 

30  . . 

9938 

7.82 

99.08 

9-54 

98.72 

11.25 

98.29 

12.94 

32  . . 

1 

99-38 

7.88 

99.07 

9.60 

98.71 

11.30 

98.28 

13.00 

34  • • 

99-37 

7-94 

99.06 

9.65 

98.69 

11.36 

98.27 

13.05 

36  . . 

99-36 

7-99 

99.05 

9.71 

98.68 

11.42 

98.25 

13.11 

38  . . 

99-35 

8.05 

99.04 

9-77 

98.67 

11.47 

98.24 

13-17 

40  . . 

99-34 

8.1 1 

99-03 

9-83 

98.65 

11-53 

98.22 

13.22 

42.  . . 

99-33 

8.17 

99.01 

9.88 

98.64 

11-59 

98.20 

13.28 

44  • • 

99-32 

8.22 

99.00 

9-94 

98.63 

1 1.64 

98.19 

13-33 

46  . . 

99-31 

8.28 

98.99 

10.00 

98.61 

11.70 

98.17 

13-39 

48  . . 

99-30 

8.34 

98.98 

10.05 

98.60 

11.76 

98.16 

13-45 

50  . . 

99.29 

8.40 

98.97 

10. 1 1 

98.58 

11.81 

98.14 

13-50 

52  . . 

99.28 

8.45 

98.96 

10.17 

98.57 

11.87 

98.13 

13-56 

54  . ^ 

99.27 

8.51 

98.94 

10.22 

98.56 

11-93 

98.11 

13.61 

56  . . 

99.26 

8.57 

98.93 

10.28 

98-54 

11.98 

98.10 

13-67 

58  . . 

99-25 

8.63 

98.92 

10.34 

98.53 

12.04 

98.08 

13-7.3 

60  , . 

99-24 

8.68 

98.91 

10.40 

98.51 

12.10 

98.06 

13-78 

<r  = 0.75 

0.75 

0.06 

0.75 

0.07 

0.75 

0.08 

0.74 

0.10 

^ = 1 .00 

1. 00 

0.08 

0.99 

0.09 

0.99 

O.II 

0.99 

0.13 

C — 1.25 

1.25 

O.IO 

1.24 

O.II 

1.24 

0.14 

1.24 

0.16 

642 


SU/^  VE  YING. 


TABLE  V. — Continued. 


Horizontal  Disiances  and  Elevations  from  Stadia  Readings. 


Minutes. 

8® 

0® 

10® 



1 1® 

Hor. 

DilT. 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff 

Dist. 

Kiev. 

Dist. 

Kiev, 

Dist. 

Kiev. 

Dist. 

Kiev. 

0 . . 

98 .06 

13-78 

97-55 

15-45 

96.98 

17.10 

96.36 

18.73 

2 . . 

98.05 

13.84 

97-53 

15-51 

96.96 

17.16 

96.34 

18.78 

4 • • 

98.03 

13.89 

97-52 

15-56 

96.94 

17.21 

1 96.32 

18.84 

6 . . 

98.01 

13-95 

97-50 

15.62 

96.92 

17.26 

1 96.29 

18.89 

'8  . . 

98.00 

14.01 

97.48 

15-67 

96.90 

17-32 

1 96.27 

18.95 

10  . . 

97.98 

14.06 

97.46 

15-73 

96.88 

17-37 

1 96-25 

19.00 

12  . . 

97-97 

14.12 

97-44 

15-78 

96.86 

17-43 

96.23 

19.05 

14  . . 

97-95 

14.17 

97-43 

15.84 

96.84 

17.48 

96.21 

19.  II 

16  . . 

97-93 

14.23 

9741 

15-89 

96.82 

17-54 

96.18 

19.16 

18  . . 

97.92 

14. 28 

97-39 

15-95 

96.80 

17-59 

96.16 

19.21 

20  . . 

97.90 

14-34 

97-37 

16.00 

96.78 

17-65 

96.14 

19.27 

22 

97.88 

14.40 

97-35 

16.06 

96.76 

17-70 

96.12 

19.32 

24  . . 

97.87 

14.45 

97-33 

1 6. 1 1 

96-74 

17-76 

96.09 

19.38 

26  . . 

97.85 

14  51 

97-31 

16.17 

96.72 

17.81 

96.07 

19-43 

28  . . 

97-83 

14.56 

97-29 

16.22 

96.70 

17.86 

96.05 

19.48 

30  . . 

97.82 

14.62 

97.28 

16.28 

96.68 

17.92 

96.03 

19.54 

32  . . 

97.80 

14.67 

97.26 

16-33 

96.66 

17.97 

96.00 

19-59 

34  . . 

97-78 

14-73 

97-24 

16.39 

96.64 

18.03 

95-98 

19.64 

36  . . 

97.76 

14.79 

97.22 

16.44 

96.62 

18.08  i 

1 95.96 

19.70 

38  . . 

97-75 

14.84 

97-20 

16.50 

96.60 

18.14 

95-93 

19-75 

40  . . 

97-73 

14.90 

97.18 

16.55 

96.57 

18.19 

95.91 

19.80 

42  . . 

97-71 

14-95 

97.16 

16.61 

96.55 

18.24 

95.89 

19.86 

44  • • 

97.69 

15.01 

97-14 

16.66 

96-53 

18.30 

95.86 

19.91 

46  . . 

97.6S 

15.06 

97-12 

16.72 

96.51 

18.35 

95.84 

19.96 

48  . . 

97.66 

15.12 

97-10 

16.77 

96.49 

18.41 

95.82 

20.02 

50  . . 

97.64 

15-17 

97.08 

16.83 

96.47 

18.46 

95-79 

20.07 

52  . . 

97.62 

15-23 

97.06 

16.88 

96.45 

18.51 

95-77 

20.12 

54  • • 

97.61 

15.28 

97-04 

16.94 

96.42 

18.57 

95-75 

20.18 

56  . . 

97-59 

15-34 

97.02 

16.99 

96.40 

18.62 

95.72 

20.23 

58  . . 

97-57 

15.40 

97.00 

17.05 

96.38 

18.68 

95-70 

20.28 

60  . . 

97-55 

15-45 

96.98 

17.10 

96.36 

18.73 

95.68 

20.34 

f = 0.75 

0.74 

O.I  I 

0.74 

0.12 

0.74 

0.14 

0.73 

0.15 

c — 1 .00 

0-99 

0.15 

0-99 

0.16 

0.98 

0.18 

0.98 

0.20 

c ■=.  1.25 

1.23 

0.18 

1.23 

0.21 

1.23 

0.23 

1.22 

0.25 

TABLES. 


643 


TABLE  V. — Continued, 

Horizontal  Distances  and  Elevations  from  Stadia  Readings. 


Minutes. 

100 

130 

140 

15° 

Hor. 

Dist. 

Diff. 

Elev. 

Hor. 

Dist. 

Diff. 

Elev. 

Hor. 

Dist. 

Diff. 

Elev. 

Hor.' 

Dist. 

Diff. 

Elev. 

0 . . 

95.6S 

20.34 

94-94 

21.92 

94-15 

23-47 

93-30 

25.00 

2 . . 

95-65 

20.39 

94.91 

21.97 

94.12 

23-52 

93-27 

25.05 

4 • • 

95-63 

20.44 

94.89 

22.02 

94-09 

23.58 

93-24 

25.10 

6 . . 

95.61 

20.50 

94.86 

22.08 

94.07 

• 23.63 

93-21 

25-15 

8 . . 

95-58 

20.55 

94.84 

22.13 

94-04 

23.68 

93.18 

25.20 

10  . . 

95-56 

20.60 

94.81 

22.18 

94.01 

23-73 

93.16 

25-25 

12  . . 

95-53 

20.66 

94-79 

22.23 

93-98 

23-78 

93-13 

25-30 

14  . . 

95-51 

20.71 

94.76 

22.28 

93-95 

23-83 

93-10 

25-35 

16  . . 

95-49 

20.76 

94-73 

22.34 

93-93 

23.88 

9307 

25-40 

18  . . 

95-46 

20.81 

9471 

22.39 

93-90 

23-93 

93-04 

25-45 

20  . . 

95-44 

20.87 

94.68 

22.44 

93-87 

23-99 

93.01 

25-50 

22  , . 

95-41 

20.92 

94.66 

22.49 

93-84 

24.04 

92.98 

25-55 

24  . . 

95-39 

20.97 

94-63 

22.54 

93.81 

24.09 

92.95 

25.60 

26  . . 

95-36 

21.03 

94.60 

22.60 

93-79 

24.14 

92.92 

25-65 

28  . . 

95-34 

21.08 

94.58 

22.65 

93-76 

24.19 

92.89 

25-70 

30  . . 

95-32 

21.13 

94-55 

22.70 

93-73 

24.24 

92.86 

25-75 

32  . . 

95-29 

21.18 

94-52 

22.75 

93-70 

24.29 

92.83 

25.80 

34  . • 

95-27 

21.24 

94-50 

22.80 

93-67 

24-34 

92.80 

25-85 

36  . . 

95-24 

21.29 

94-47 

22.85 

93-65 

24-39 

92.77 

25.90 

38  . . 

95.22 

21.34 

94-44 

22.91 

93-62 

24.44 

92.74 

25-95 

40  . . 

95-19 

21.39 

94.42 

22.96 

93-59 

24-49 

92.71 

26.00 

42  . . 

95-17 

21.45 

94-39 

23.01 

93-56 

24-55 

92.68 

26.05 

44  • • 

95-M 

21.50 

94-36 

23.06 

93-53 

24.60 

92.65 

26.10 

46  . . 

95.12 

21-55 

94-34 

23.11 

93-50 

24.65 

92.62 

26.15 

48  . . 

95-09 

21.60 

94-31 

23.16 

93-47 

24.70 

92.59 

26.20 

50  . . 

95-07 

21.66 

94.28 

23.22 

93-45 

24-75 

92.56 

26.25 

52  . . 

95-04 

21.71 

94.26 

23.27 

93-42 

24.80 

92.53 

26.30 

54  • • 

95.02 

21.76 

94-23 

23-32 

93-39 

24.85 

92.49 

26.35 

56  . . 

94-99 

21.81 

94.20 

23-37 

93-36 

24.90 

92.46 

26.40 

58  . . 

94-97 

21.87 

94.17 

23.42  * 

93-33 

24-95 

92.43 

26.45 

60  . . 

94.94 

21.92 

94.15 

23-47 

93-30 

25.00 

92.40 

26.50 

-^  = 0.75 

0.73 

0.16 

0-73 

0.17 

0.73 

0.19 

0.72 

0.20 

c — I.OO 

0.98 

0.22 

0.97 

0.23 

0.97 

0.25 

0.96 

0.27 

c = 1.25 

1.22 

0.27 

1. 21 

0.29 

1. 21 

0.31 

1.20 

0.34 

644 


SUR  VE  YING. 


TABLE  V.  — Continued. 


Horizontal  Distances  and  Elevations  from  Stadia  Readings. 


Minuten. 

170 

18° 

11>° 

Hor. 

Dinr. 

Hor. 

Diir. 

Hor. 

Iiiff. 

Hor. 

DifT. 

Dist. 

Elev. 

Dist. 

Elcv. 

Dist. 

Elcv. 

Dist. 

Elcv. 

0 . . 

92.40 

26.50 

91.45 

27.96 

90-15 

29.39  ' 

89.40 

30.78 

2 . . 

92-37 

26.55 

91.42 

28.01 

90.42 

29.44  ' 

89.36  1 

30-83 

4 • • 

92-34 

26.59 

9 '-39 

28.06 

90.38 

29.48 

89-33 

30.87 

C . . 

92.31 

26.64 

91-35 

28.10 

90-35 

29-53 

89.29 

30.92 

8 . . 

92.28 

26.69 

91.32 

28.15 

90.31 

29-58 

89.26 

30-97 

10  . . 

92.25 

26.74 

91.29 

28.20 

90.28 

29.62 

89.22 

31.01 

12  . . 

92.22 

26.79 

91.26 

28.25 

90.24 

29.67 

89.18 

31.06 

14  . . 

92.19 

26.84 

91.22 

28.30 

90.21 

29-72 

89.15 

31.10 

i6  . . 

92.15 

26.89 

91.19 

28.34 

90.18 

29.76 

89.1 1 

3i-'S 

i8  . . 

92.12 

26.94 

91.16 

28.39 

90.14 

29.81 

89.08 

31-19 

20  . . 

92.09 

26.99 

91.12 

28.44 

90.1 1 

29.86 

89.04 

3'-24 

22  . . 

92.06 

1 

27.04 

91.09 

28.49 

90.07 

29.90 

89.00 

31.28 

24  . . 

92.03 

27.09 

91.06 

28.54 

90.04 

29.95 

88.96 

3''33 

26  . . 

92.00 

27-13 

91.02 

28.58 

90.00 

30.00 

88.93 

31-38 

28  . . 

91.97 

27.18 

90.99 

28.63 

89.97 

30.04 

88.89 

31.42 

30  . . 

91-93 

27.23 

90.96 

28.68 

89.93 

30.09 

88.86 

3'-47 

32  . . 

91.90 

27.28 

90.92 

28.73 

89.90 

30.14 

88.82 

31-5' 

34  . • 

91.87 

27-33 

90.89 

28.77 

89.86 

30.19 

88.78 

3'-56 

36  . . 

91.84 

27.38^ 

90.86 

28.82 

89.83 

30-23 

88.75 

31.60 

38  . . 

91.81 

27-43' 

90.82 

28.87 

89.79 

30.28 

88.71 

31-65 

40  . . 

91.77 

27.48 

90.79 

28.92 

89.76 

30-32 

88.67 

31.69 

42  : . 

91.74 

27-52 

90.76 

28.96 

89.72 

30-37 

88.64 

3'-74 

44  • • 

91.71 

27.57 

90.72 

29.01 

89.69 

30.41 

88.60 

31-78 

46  . . 

91.68 

27.62 

90.69 

29.06 

89.65 

30.46 

88.56 

31-83 

48  . . 

91.65 

27.67 

90.66 

29.1 1 

89.61 

30-51 

88.53 

3 '-87 

50  . . 

91.61 

27.72 

90.62 

29.15 

89.58 

30-55 

88.49 

31.92 

52  . . 

91.58 

27-77 

90-59 

29.20 

89-54 

30.60 

88.45 

31.96 

54  . • 

9t-55 

27.81 

90-55 

29-25 

89.51 

30-65 

88.41 

32-01 

56  . . 

91.52 

27.S6 

90.52 

29-30 

89-47 

30.69 

88.38 

32-05 

58  . . 

91.48 

27.91 

90.48 

29-34 

89.44 

30-74 

88.34 

32.09 

60  . . 

91-45 

27.96 

90-45 

29-39 

89.40 

30.78 

88.30 

32-14 

^ = 075 

0.72 

0.21 

0.72 

0.23 

0.71 

0.24 

0.71 

0.25 

c = 1 .00 

0.86 

0.28 

0.95 

0.30 

0-95 

0.32 

0.94 

0-33 

«rz=  1.25 

1.20 

0-35 

1. 19 

0.38 

1. 19 

0.40 

1. 18 

0.42 

TABLES. 


645 


TABLE  V. — Contintud. 


Horizontal  Distances  and  Elevations  from  Stadia  Readings. 


Minutes. 

0 

0 

21° 

22° 

23° 

Hor. 

Dist. 

Diff. 

Elev. 

Hor. 

Dist. 

Diff. 

Elev. 

Hor. 

Dist. 

Diff. 

Elev. 

Hor. 

Dist. 

Diff 

Elev. 

0 . . 

88.30 

32.14 

87.16 

33-46 

85-97 

34-73 

84-73 

35-97 

2 . . 

88.26 

32.18 

87.12 

33-50 

85-93 

34-77 

84.69 

36.01 

4 • • 

88.23 

32.23 

87.08 

33-54 

85.89 

34.82 

84.65 

36.05 

6 . . 

88.19 

32.27 

87.04 

33-59 

85.85 

34.86 

84.61 

36.09 

8 . . 

88.15 

32.32 

87.00 

33-63 

85.80 

34-90 

84-57 

36-13 

10  . . 

88.11 

32.36 

86.96 

' 33-67 

85-76 

34.94 

84.52 

36.17 

12  . . 

88.08 

32.41 

86.92 

33-72 

85.72 

34.98 

84.48 

36.21 

14  . . 

88.04 

32.45 

86.88 

33-76 

85.68 

35-02 

84-44 

36.25 

16  . . 

88.00 

32-49 

86.84 

33-80 

85.64 

35-07 

84.40 

36.29 

18  . . 

87.96 

32.54 

86.80 

33-84 

85.60 

35-11 

84.35 

36-33 

20  . . 

87-93 

32.58 

86.77 

33-89 

85.56 

35-15 

84.31 

36.37 

22  . . 

87.89 

32-63 

86.73 

33-93 

85.52 

35-19 

84.27 

36.41 

24  . . 

87.85 

32.67 

86.69 

33-97 

85.48 

35-23 

84.23 

36.45 

26  . . 

87.81 

32-72 

86.65 

34.01 

85-44 

35-27 

84.18 

36.49 

28  . . 

87.77 

32.76 

86.61 

34-06 

8540 

35-31 

84.14 

36.53 

30  . . 

87.74 

32.80 

86.57 

34.10 

85.36 

35-36 

84.10 

36.57 

32  . . 

87.70 

32.85 

86.53 

34-14 

85.31 

35-40 

84.06 

36.61 

34  • . 

87.66 

32.89 

86.49 

34.18 

85.27 

35-44 

84.01 

36-65 

36  . . 

87.62 

32-93 

86.45 

34-23 

85-23 

35-48 

83-97 

36.69 

38  . . 

87.58 

32.98 

86.41 

34-27 

85.19 

35-52 

83-93 

36.73 

40  . . 

87.54 

33-02 

86.37 

34-31 

85-15 

35-56 

83.89 

36.77 

42  . . 

87-51 

33-07 

86.33 

34-35 

85.11 

35-60 

83.84 

36.80 

44  • • 

87.47 

33-11 

86.29 

34-40 

85-07 

35-64 

83.80 

36.84 

46  . . 

87-43 

33-15 

86.25 

34-44 

85.02 

35.68 

83.76 

36.88 

48  . . 

87-39 

33-20 

86.21 

34-48 

84.98 

3572 

83-72 

36.92 

50  . . 

87-35 

33.24 

86.17 

34-52 

84-94 

35-76 

83.67 

36.96 

52  . . 

87-31 

C..J 

to 

00 

86.13 

34-57 

84.90 

35-80 

83-63 

37-00 

54  • • 

87.27 

33-33 

86.09 

34.61 

84.86 

35-85 

83-59 

37-04 

56  . . 

87.24 

33-37 

86.05 

34-65 

84.82 

35-89 

83-54 

37-08 

58  . . 

87.20 

33.41 

86.01 

34-69 

84-77 

35-93 

83-50 

37.12 

Co  . . 

87.16 

33.46 

85-97 

34-73 

84-73 

35-97 

83-46 

37-16 

<r  = 0.7S 

0.70 

0.26 

0.70 

0.27 

0.69 

0.29 

0.69 

0.30 

f = 1 .00 

0.94 

0.35 

0-93 

0.37 

0.92 

0.38 

0.92 

0.40 

c = 1.25 

1. 17 

0.44 

1. 16 

0.46 

1. 15 

0.48 

1. 15 

0.50 

646 


SURVEYING. 


TABLE  V.  — Continued. 


Horizontal  Distances  and  Elevations  from  Stadia  Readings. 


, 

Minutes. 

24° 

25° 

20° 

27° 

Hor. 

DifT. 

Hor. 

DilT. 

Hor. 

Diff. 

Hor. 

DifT. 

Dist 

Elcv. 

Dist. 

Elcv 

Dist, 

Elcv. 

Dist. 

Elcv. 

0 . . 

83.46 

37-i6 

82.14 

38.30 

80.78 

39-40 

79-39 

40.45 

2 . . 

83.41 

37.20 

82.09 

38.34 

80.74 

39-44 

79-34 

40.49 

4 • • 

83-37 

37-23 

82.05 

38.38 

80.69 

39-47 

79-30 

40.52 

6 . . 

83-33 

37-27 

82.01 

38.41 

80.65 

39-51 

79-25 

40.55 

8 . . 

83.28 

37-31 

81.96 

38.45 

80.60 

39-54 

79.20 

40.59 

10  . . 

83.24 

37-35 

81.92 

38.49 

80.55 

39-58 

79-15 

40.62 

12  . . 

83.20 

37-39 

81.87 

38.53 

80.51 

39.61 

79.11 

40.66 

14  . . 

83-15 

37-43 

81.83 

38.56 

80.46 

39-65 

79.06 

40.69 

16  . . 

83.11 

37-47 

81.78 

38.60 

80,41 

39-69 

79.01 

40.72 

18  . . 

83.07 

37-51 

81.74 

38.64 

80.37 

39-72 

78.96 

40.76 

20  . . 

83.02 

37-54 

81.69 

38.67 

80.32 

39-76 

78.92 

40.79 

22  . . 

82.98 

37.58 

81.65 

38.71 

80.28 

39-79 

78.87 

40.82 

24  . . 

82.93 

37.62 

81.60 

38.75 

80.23 

3983 

78.82 

40.86 

26  . . 

82.89 

37.66 

81.56 

38.78 

80.18 

39.86 

78.77 

40.89 

28  . . 

82.85 

37-70 

81.51 

38.62 

80.14 

39-90 

78.73 

40.92 

30  • • 

82.S0 

37-74 

81.47 

38.86 

80.09 

39-93 

78.68 

40.96 

32  . . 

82.76 

37-77 

00 

L 

38.89 

80.04 

39-97 

78.63 

40.99 

34  • . 

82.72 

37-8i 

81.38 

38.93 

80.00 

40.00 

78.58 

41.02 

36  . . 

82.67 

37.85 

81.33 

38.97 

79-95 

40.04 

78.54 

41.06 

38  . . 

82.63 

37-89 

81.28 

39.00 

79-90 

40.07 

78.49 

41.09 

40  . . 

82.58 

37.93 

81.24 

39-04 

79.86 

40.11 

78.44 

41.12 

42  . . 

82.54 

37-96 

81.19 

39.0S 

7981 

40.14 

78.39 

41.16 

44  • • 

82.49 

38.00 

81.15 

39-11 

7976 

40.18 

78.34 

41.19 

46  . . 

82.45 

38.04 

81.10 

39- 15 

79.72 

40.21 

78.30 

41.22 

48  . . 

82.41 

38.08 

81.06 

39.18 

79.67 

40.24 

78.25 

41.26 

50  . . 

82.36 

38.11 

81.01 

39.22 

79.62 

40,28 

78.20 

41.29 

52  . . 

82.32 

38.15 

80.97 

39.26 

79.58 

40.31 

78.15 

41.32 

54  . . 

82.27 

38.19 

80.92 

39-29 

79-53 

40.35 

78.10 

41.35 

56  . . 

82.23 

38.23 

80.87 

39-33 

79.48 

40.38 

78.06 

41.39 

58  . . 

82.18 

38.26 

80.83 

39-36 

79-44 

40.42 

78.01 

41.42 

60  . . 

82.14 

38.30 

80.78 

39-40 

79-39 

40.45 

77-96 

41.45 

cz=o.7S 

0.68 

0.31 

0.68 

0.32 

0.67 

0.33 

0,66 

0.35 

Czzil.oo 

0.91 

0.41 

0.90 

0.43 

0.89 

0.45 

0.89 

0.46 

C=1.25 

1. 14 

0.52 

113 

0.54 

1. 12 

0.56 

I. II 

0.58 

TABLES. 


647 


TABLE  V. — Contimied. 


Horizontal  Distances  and  Elevations  from  Stadia  Readings. 


Minutes. 

28° 

29° 

30° 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Dinr. 

Dist. 

Elev. 

Dist 

Elev. 

Dist. 

Elev. 

0 . . 

77.96 

41-45 

76.50 

42.40 

75-00 

43-30 

2 . . 

77.91 

41.48 

76.45 

4 2. .43 

74-95 

43-33 

4 . • 

77.86 

41.52 

76.40 

42.46 

74-90 

43-36 

6 . . 

77.81 

41-55 

76.35 

42.49 

74-85 

43-39 

8 . . 

7777 

41.58 

76.30 

42.53 

74.80 

43-42 

10  . . 

77.72 

41.61 

76.25 

42.56 

74-75 

43-45 

12  . . 

77.67 

41.65 

76.20 

42.59 

74-70 

43-47 

14  . . 

77.62 

41.68 

76.15 

42.62 

74-65 

43-50 

16  . . 

77-57 

41.71 

76.10 

42.65 

74.60 

43-53 

18  . . 

77-52 

41.74 

76.05 

42.68 

74-55 

43-56 

20  . . 

77.48 

41-77 

76.00 

42.71 

74-49 

43-59 

22  . . 

77.42 

41.81 

75-95 

42.74 

74-44 

43.62 

24  . . 

77-38 

41.84 

75-90 

42.77 

74-39 

43-65 

26  . . 

77-33 

41.87 

75-85 

42.80 

74-34 

43-67 

28  . . 

77.28 

41.90 

75.80 

42.83 

74.29 

43-70 

30  . . 

77-23 

41-93 

75-75 

42.86 

74-24 

43-73 

32  . . 

77.18 

41.97 

75-70 

42.89 

74.19 

43-76 

34  . • 

77-13 

42.00 

75-65 

42.92 

74.14 

43-79 

36  . . 

77-09 

42.03 

75.60 

42.95 

74-09 

43.82 

38  . . 

77.04 

42.06 

75-55 

42.98 

74-04 

43-84 

40  . . 

76.99 

42.09 

75-50 

43.01 

73-99 

43-87 

42  . . 

76.94 

42.12 

75-45 

43-04 

73-93 

43-90 

44  • • 

76.89 

42.15 

7540 

43-07 

73-88 

43-93 

46  . . 

76.84 

42.19 

75-35 

43.10 

73-83 

43-95 

48  . . 

76.79 

42.22 

75-30 

43-13 

73-78 

43-98 

50  . . 

76.74 

42.25 

75-25 

43.16 

73-73 

44.01 

52  . . 

76.69 

42.28 

75.20 

43.18 

73-68 

44-04 

54  . . 

76.64 

42.31 

75-15 

43.21 

73-63 

44-07 

56  . . 

76.59 

42.34 

75.10 

43-24 

73-58 

44-09 

58  . . 

76.55 

42.37 

75-05 

43-27 

73-52 

44.12 

60  . . 

76.50 

42.40 

75.00 

43-30 

73-47 

44-15 

^ = 0.75 

0.66 

0.36 

0.65 

1 0.37 

0.65 

0.38 

I.OO 

0.88 

0.48 

0.87 

0.49 

0.86 

0.51 

c ~ 1.25 

1. 10 

0.60 

1.09 

0.62 

1.08 

0.64 

648 


SURVEYING. 


TABLE  VI. 

Natural  Sines  and  Cosines. 


0° 

1 

0 

2° 

3 

4 

1 

Rine 

Cosin 

Sine 

Cosin 

1 Sine  1 

Cosin 

Sine  1 

Cosin 

Sine  1 

Cosin  1 

0 

.00000 

Cue. 

.01745 

.9998.5 

.03190' 

. 999:J9 

To.523  4 

.9980.3 

.oooro 

.997.56 

60 

1 

.(10029 

One. 

.01774 

.9998.4 

.0:1.519 

.999:38 

.0.5‘26:31 

.99801 

.()7(K>5 

.997.54 

59 

2 

.00058 

One. 

.01803 

.9998.4 

.0:3.518, 

.999:57 

.05292 

.99800 

.070:41 

.997.52 

58 

3 

.00087 

One. 

.018.32 

.99983 

.0:3.577! 

.999:56 

.05.321 

.998.58 

.(OOO-l 

.9!)7.5<) 

57 

4 

.00116 

One. 

.01862 

.9998:3 

.0:3606: 

.999:35 

.0.53.50! 

.998.57 

.07(K)2 

.99748 

56 

6 

.00145 

One. 

.01891 

.99982 

.0.36:35 

.999:5-4 

.05:379, 

.998.55 

.07121 

.99746 

55 

C 

.00175 

One. 

.01920 

.99982 

.0.3664! 

.999:3.3 

.054081 

.998.51 

.071.51 

.99744 

.54 

7 

.00204 

One. 

.01949 

.99981 

.0:3693 

.99932 

.0.31.37' 

.998.52 

.07179 

.99742 

53 

8 

.00233 

One. 

.01978 

.99980 

.0:372:3 

.99931 

.0.5466! 

.99851 

: .07208 

.99740 

52 

y 

.00262 

One. 

.02007 

.99980 

.0.37.52 

.999:30 

.0.319.5I 

.99.849 

.072371 

[.997.38 

51 

10 

.00291 

One. 

.02036 

.99979 

.03781 

.99929 

.05524 1 

.9984i 

.07266 

.99736 

50 

11 

.00320 

.99909 

' .02065 

.99970 

.03810 

.90927 

.0.5.5.53 

.99846' 

.07295 

.997.34 

49 

12 

.00.349 

. 99999 

.02094 

.99978 

.0:3839 

.99926 

.0.5.582 

.99844 

.07:121 

.997.31 

48 

13 

.00378 

.99999 

.02123 

.999771 

.0.3868 

.99925 

.0.5611 

.99842 

.07.3.5.3 

.99729 

47 

14 

.00407 

.99999 

.021.52 

.999171 

.0.3897 

.99924 

.0.5640 

.99841' 

.07:182 

.99727 

46 

15 

.00436 

.99999 

.02181 

.99976' 

.0.3926 

.99923 

.05669 

.99839 

.07411 

.9972.5 

45 

IG 

.00465 

.02211 

.99976 

' .0:39.5.5 

.99922 

.05698 

.99838 

.07440 

.9972:1 

44 

17 

.00495 

.99999 

.02;240 

.99975 

.0.3984 

.99921 

.05727 

.99836 

.07469 

.99721 

4.3 

18 

.00.524 

.99999 

.02269 

.99974 

1.04013 

.99919 

.0.57.56 

.99834 

.074;i8 

.99719 

42 

19 

.00.553 

.99998 

.02298 

.99974 

1.04042 

.99918' 

.0.5785 

.9983.3 

.07.527 

.99716 

41 

20 

.00582 

.02:327 

.99973 

1.04071 

.99917, 

.05814 

.99831 

.07556 

.99714 

140 

21 

.00611 

.99998 

,02.356 

.99972 

1.04100 

.99916 

.0.5844 

.99829 

.07.5a5 

.99712 

I39 

22 

.00640 

.99998 

.02385 

.99972 

.04129 

.99915 

.05873 

.99827 

.07614 

.99710 

! 38 

23 

.00669 

.99998 

.02414 

.90971 

.041.59 

.99913 

.0.5902 

.99826 

.(■7643 

.99708 

37 

24 

.00698 

OQdOU 

.02443 

.999701 

.04188 

.99912 

.0.5931 

.99824 

.07672 

.99705 

36 

25 

.00727 

.99997 

.02472 

.999691 

.04217 

.99911 

.0.5960 

.99822 

.07701 

.99703 

35 

2G 

.00756 

.99997, 

.02.501 

.999691 

.04246 

.99910 

.05989 

.99821 

.077.30 

.99701 

34 

27 

.00785 

.99997 

.02530 

.99968 

.04275 

.99909 

.06018 

.99819 

.07759 

.99699 

33 

28 

00814 

.99997 

.02560 

.99967 

.01.304 

.99907 

! .06047 

.99817 

.07788 

.99696 

1 32 

29 

.00844 

.99996 

i .02.589 

1.99966 

.04.3.33 

.99906 

' .06076 

.99815 

.07817 

.99694 

1 31 

30 

.00873 

,.99996 

.02618 

.99966 

1.04362 

.99905 

1 .06105 

.99813 

.07846 

.99692 

30 

31 

.00902 

L 99996 

.02647 

.99965 

.04391 

.99904 

! .06134 

.99812 

.07875 

.99689 

29 

32 

.00931 

1.99996 

! .02676 

.90964 

.04420 

.99902 

' .06163 

.99810 

.07904 

.99687 

28 

33 

.00960 

.99995 

.02705 

.99963 

.04449 

.99901 

1 06192 

.99808 

.079.33 

.99685 

27 

34 

.00989 

.99995 

.02734 

.99963 

.04478 

.99900 

! .06221 

.99806 

' .07962 

.996a3 

26 

35 

.01018 

.99995 

.02763 

.99902 

.04507 

.99898 

1 .06250 

.99804 

: .07991 

.99680 

25 

36 

.01047 

.99995 

.02792 

.99961 

.04536 

.99897 

1 .06279 

.99803 

.08020 

.99678 

24 

37 

.01076 

.99994 

.02821 

.99960 

.04565 

.99896 

.06.308 

.99801 

.08049 

.99676 

23 

38 

.01105 

.99994 

.02850 

.99959 

.04594 

.99894 

! .06337 

.99799 

.08078 

1.99673 

22 

39 

.01134 

.99994 

.02879 

.99959 

.04623 

.99893 

1 .06366 

.99797 

.08107 

1.99671 

21 

40 

.01164 

.99993 

.02908 

.99958 

.04653 

.99892 

1 .06395 

.99795^ 

.08136 

.99668 

20 

41 

.01193 

.99993 

.02938 

.99957 

.04682 

.99890 

[.06424 

.99793' 

.08165 

.99666 

19 

42 

.01222 

.99993 

.02967 

.99956 

.04711 

.99889 

' .06453 

.99792 

.08194 

.99664 

18 

43 

.01251 

.99992 

.02996 

.99955 

.04740 

.99888 

[ .06482 

.99790 

.08223 

.99661 

17 

44 

.01280 

.99992 

.03025 

.99954 

.04769 

1.99886 

.06511 

.99788 

.08252 

.99659 

16 

45 

.01309 

.99991 

.0:3054 

.99953 

.04798 

.99885 

.06540 

.99786 

.08281 

.99657 

15 

46 

.01338 

.99991 

.03083 

.99952 

.04827 

.99883 

.06569 

.99784 

.08310 

.99654 

14 

47 

.01367 

.99991 

.03112 

.99952 

.04856 

.99882 

.06598 

.99782 

.O&3.39 

.99652 

13 

48 

.01396 

.99990 

.03141 

.99951 

.04885 

.99881 

.06627 

.99780 

.08368 

.99649 

12 

49 

.01425 

.99990 

.03170 

.99950 

[.04914 

.99879 

.06656 

.99778 

.0^397 

.99647 

11 

50 

.01454 

.99989 

.03199 

.99949 

.04943 

.99878 

.06685 

.99776 

.0&42G 

.99644 

10 

51 

.01483 

.99989 

.03228 

.99948 

.04972 

.99876 

.06714 

.99774 

.08455 

.99642 

9 

52 

.01513 

.99989 

.032.57 

.99947 

.05001 

.99875 

.06743 

.99772 

.08484 

.99639 

8 

53 

.01542 

.99988 

.03286 

.99946 

.0.5030 

.99873 

.06773 

.99770' 

.08513 

1.99637 

7 

54 

,01571 

.99988 

.03316 

.99945 

.0.50.59 

.99872 

.06802 

.99768- 

.08542 

.99635 

6 

55 

.01600 

.99987 

.03.345 

.99944 

.0.5088 

.99870 

.06831 

.997661 

.08571 

.99632 

5 

56 

.01629 

.99987 

.0:3:374 

.99943 

.05117 

.99869 

.06860 

.997641 

.08600 

.99630 

4 

57 

.01658 

.99986 

.0:3403 

.99942 

.05146 

.99867 

.06889 

.997621 

.08629 

.99627 

3 

58 

.01687 

.99986 

.0:34:32 

.99941 

.05175 

.99866 

.06918 

.99760! 

.08658 

.99625 

2 

59 

.01716 

.99985 

.0.3461 

.99910 

.0.5205 

.99864 

.06947 

.99758! 

.08687 

.99622 

1 

60 

.01745 

.99985 

.0.3490 

.99939 

.0.5234 

.99863 

.06976 

.997.56 

.08716 

.99619 

/ 

Cosin 

1 Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

0 

0 

M 

B 

1 Sine 

/ 

89“ 

0 

00 

00 

87“ 

86“ 

0 

m 

GO 

TABLES. 


649 


TABLE  VI. — Continued . 

^ Natural  Sines  and  Cosines. 


5 

0 

6 

7 

» 

8 

9 

9 

0 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosln 

Sine 

Cosin 

Sine 

Cosin 

/ 

0 

.08716 

.99619 

. 10453 

.99452 

.12187 

.992M 

.13917 

T99CW 

.15643 

. 98709 

1 

.08745 

.99617 

.10482 

.99449 

.12216 

.99251 

.13946 

.99023 

.15672 

.987641 

59 

2 

.08774 

.99614 

.10511 

.99446 

.12245 

.99248 

.13975 

.99019 

’ .1.5701 

.98760 

58 

3 

.08803 

.99612 

.10540 

.99443 

.12274 

.99244 

.14004 

.99015 

.15730 

.98755! 

57 

4 

.08831 

.99609 

.10,569 

.99440 

.12:302 

.99240 

.140.33 

.99011 

.157'58 

.98751 

56 

5 

.08860 

.99607 

.10597 

.994:37 

.12331 

.992.37 

.14061 

.99000 

.15787 

.98746: 

55 

6 

.08889 

.99604 

.10626 

.994:341 

.12360 

.99233; 

.14090 

.99002 

.15816 

.98741! 

.54 

7 

.08918 

99602 

.10655 

.994:31 

.12389 

.99230 

.14119 

.98998 

.15845 

.987.371 

53 

8 

.08947 

.99599 

.10684 

.99428' 

.12418 

.99226' 

.14148 

.98994 

.1.5873 

GO 

52 

9 

.08976 

.99596 

.10713 

.994241 

.124471.992221 

.14177 

.98990 

.15902 

.98728 

51 

10 

.09005 

.99594 

.10742 

.99421 1 

.12476 

.99219 

.14205 

.98986 

j .15931 

.98723 

50 

11 

.09034 

.99591 

.10771 

.99413! 

.12504 

.99215 

.14234 

.98982 

! .15959 

.98718 

49 

12 

.09063 

.99588 

.10800 

.99415 

.12533 

.99211 

.14263 

.98978 

i .15988 

.98714 

48 

13 

.09092 

.99586 

.10829 

.99412' 

.12562 

.99208 

.14292 

.98973 

! .16017 

.98709 

47 

14 

.09121 

.99583 

. 10858 

.994091 

.12591 

.99204 

.14320 

.98969 

.16046 

.98704 

46 

15 

.09150 

.99580 

.10887 

.99406 

.12620 

.99200 

.14349 

.98965 

.16074 

.98700' 

45 

16 

.09179 

.99578 

.10916 

.99402 

.12049 

.99197 

.14378 

.98961 

.16103 

.98095' 

44 

17 

.09208 

.99575 

.10945 

.99399 

.12678 

.9919.3 

.14407 

.989.57 

.16132 

.98690 

43 

18 

.09237 

.99572 

.10973 

.99.396 

.12706 

.99189 

.14436 

.98953 

.16160 

.98680 

42 

19 

.09266 

.99570 

.11002 

.9939:3 

.12735 

.99186 

.14464 

.98948 

.16189 

.98681 

41 

20 

.09295 

.99567 

.11031 

.99390 

.12764 

.99182 

.14493 

.98944 

.16218 

. 98676 1 

40 

21 

.09324 

.99564 

.11060 

.99386 

.12793 

.99178 

.14522 

.98940 

.16246 

.98671 

39 

22 

.09353 

.99562 

.11089 

.99383 

.12822 

.99175 

.14551 

.98936 

.16275 

.98667' 

38 

23 

.09382 

.99559 

.11118 

.99380 

.12851 

.99171 

.14580 

.98931 

.16304 

.98662; 

37 

24 

.09411 

. 99556 

.11147 

.99377 

.12880 

.99107 

.14608 

.98927 

.16333 

.98657: 

36 

25 

.09440 

.99553 

.11176 

.99.374 

.12908 

.99163 

.14637 

.98923 

.16361 

.98652 

35 

26 

.09469 

.99551 

.11205 

.99370 

.129:37 

.99160 

.14666 

.98919 

.16390 

.986481 

34 

27 

.09498 

.99548 

.11234 

.99,367 

.12966 

.99156 

.14695 

.98914 

.16419 

.980431 

33 

28 

.09527 

.99545 

.11263 

.99364 

.12995 

.99152 

.1472.3 

.98910 

.10447 

.986381 

32 

29 

.09556 

.99542 

.11291 

.99,300 

.1:3024 

.99148 

.14752 

.98906 

.16476 

.98633 

31 

30 

.09585 

.99540 

.11320 

.99357 

.13053 

.99144 

.14781 

.98902 

.16505 

.98629 

30 

31 

.09614 

.99,537 

.11349 

.99354 

.13081 

.99141 

.14810 

.98897 

.16533 

.98624 

29 

32 

.09642 

.99.5:44 

.11.378 

.99:351 

.13110 

.99137 

.14838 

.98893 

.16562 

.98619 

28 

33 

.09671 

.99531 

.11407 

.99347 

.1.3139 

.99133 

.14867 

.98889 

.10591 

.98614 

27 

34 

.09700 

.99528 

.11436 

.99:344 

.13168 

.99129 

.14896 

.98884 

: .16620 

.98609 

26 

35 

.09729 

.99526 

.11465 

.99341' 

.13197 

.90125 

.14925 

1 .98880 

1 .16648 

.98604 

25 

36 

.09758 

.99523 

.11494 

.99.3.37 

.1,3226 

.99122 

.14954 

1.98876 

^ .16677 

.98600 

24 

37 

.09787 

.99520 

.11523 

.99.3.341 

.13254 

.99118 

.14982 

.98871 

, .16706 

.98595 

23 

38 

.09816 

.99517 

.11552 

.99.331 

.1.3283 

.99114 

.15011 

.98867 

i .16734 

.98590 

22 

39 

.09845 

.99.514 

.11580 

.99.327 

.13312 

.99110 

.15040 

.98863 

.16763 

.88585 

21 

40 

.09874 

.99511 

.11609 

.99324 

.13341 

.99106 

.15069 

.98858 

.16792 

.98580 

20 

.09903 

.99.508 

.11638 

.99320 

.1,3370 

.99102 

.15097 

.98854 

.16820 

.98575 

19 

42 

.09932 

.99.506 

.11667 

.993171 

.13.399 

.99098 

.15126 

.98849 

.16849 

.98570 

18 

43 

.09961 

.99503 

.11696 

.993141 

.13427 

.99094 

.15155 

.98845 

.16878 

.98565 

17 

44 

.09990 

.99500 

.11725 

.99.310 

.13456 

.99091 

.15184 

.98841 

.16906 

.98561 

16 

45 

.10019 

.99497 

.117.54 

•99307 

.13485 

.99087 

.15212 

.98836 

.16935 

.98556 

15 

46 

.10048 

.99494 

.1178:3 

.99.3031 

.13514 

.99083 

.15241 

.98832 

.16964 

.98.551 

14 

47 

.10077 

.99491 

.11812 

.99.300, 

.13543 

.99079 

.15270 

.98827 

.16992 

.98546 

13 

48 

.10106 

.99488 

.11840 

.99297 

.1.3.572 

.99075 

.15299 

.98823 

I .17021 

.98.541 

12 

49 

.10135 

.99185 

.11869 

.99293' 

.13600 

.99071 

.15327 

.98818 

.17050 

.98536 

11 

50 

.10164 

.99482 

.11898 

.99290 

.13029 

.99067 

.15356 

.98814 

.17078 

.98531 

10 

51 

.10192 

.99479 

.11927 

.99286' 

.136.58 

.9906.3 

.1.5385 

.98809 

.17107 

.98.526 

9 

52 

.10221 

.99476 

1 .119.56 

.99283 

.1.3687 

.99059 

.15414 

.98805 

.17136 

.98521 

8 

53 

1 . 10250 

.99473 

1 .11985 

.99279 

.13716 

.990.55 

.1.5442 

1.98800 

.17164 

.98516 

7 

54 

.10279 

.99470 

.12014 

.99276 

.1.3744 

.99051 

.1.5471 

.98796 

.17193 

.98511 

6 

55 

i .10308 

.99467 

.12043 

.99272 

.1.3773 

.99047 

.1.5.500 

i. 98791 

.17222 

.98.506 

5 

56 

1 .10337 

.99464 

.12071 

.99269 

.1.3802 

.99043 

.1.5.529 

i. 98787 

.17250 

.98.501 

4 

57 

10366 

,.99461 

.12100 

.99265 

.i:i83l 

.99039 

.1.5.5.57 

.98782 

.17279 

.98496 

3 

58  .10395 

.994.58 

s .12129 

.99262 

.13860 

.990.35 

.1.5.586 

.98778 

.17.308 

.98491 

2 

59 

' .10424 

.994.55 

.121.58 

.992.58 

.1.3889 

.99031 

.1.5615 

.98773 

.17:336 

.98486 

1 

60 

.10453 

,.994.52 

.12187 

. 992.55  j 

.1:191 7 

.99027 

.15643 

i. 98769 

.17.365 

.98481 

0 

/ 

Cosin 

1 Sine 

Cosin 

Sine  ! 

Cosin 

Sine 

Cosin 

1 Sine 

Cosin 

Sine 

t 

84» 

1 83“  1 

82“ 

81“ 

80“ 

650 


SUR  VE  YING. 


TABLE  VI. — Continued. 

Natural  Sines  and  Cosines. 


1 10° 

11° 

12° 

13“ 

II  H’  1 

Rino 

Cosin 

Sine 

Cosin 

' Sine 

Cosin 

Sine 

1 Cosin 

1 Sine 

(’osIn 

0 

I.173G5 

.9.8181 

.19)81 

.98]  0:3 

1 .2079i 

'.97)^) 

>2495 

■97437 

721192 

!):o:3»!  (T) 

1 

1.17393 

.98470 

.19109 

.981.57 

1 .2))M'20 

.97809 

.22.52.3 

.974:30 

11  .242-2) 

.!)7^)^•5  .50 

2 

! .17422 

.9.8471 

.I9i:}8 

.981.52 

1 .20848 

1.97803 

.22.5.52 

.97421 

.2124  - 

.07<tl  .5  .58 

3 

1.17151 

.98 100 

.19107 

.98140 

I .20877 

.97797 

.22.580 

.9ril7 

1 .21277 

.OTIHH  .57 

A 

1.17479 

.9.8401 

.19195 

.98140 

.2.)91)5 

'.97791! 

.22008 

.97411 

r .21:3)5 

.9;  0)1  .50 

5 

1 .17508 

.981.55 

.19224 

.931:3.5 

.209. 13 

.97784 

.220.37 

' 97101 

.21.333 

.!.oo'.)r  .55 

G 

1.17537 

.984.50 

.192.52 

.9  3] ‘29 

.2.0902 

.97778 

.22005 

4)7:398 

.21:502 

. 00087 : .54 

7 

.17505 

.98115 

.19281 

.931‘21 

.2')9:>() 

.97772 

.22093 

.97:i.)l 

[ .21:500 

.00080'  .53 

8 

1.17591 

.98110 

.19:309 

.93118 

.21019 

.9770() 

22722 

.97:381 

.21118 

.90!)73:  .52 

9 

.17021 

.9.8135 

.193.3,8 

.93112 

.21017 

.95700 

.227^ 

.97:578 

.21110 

, 00000 1 .51 

10 

.17051 

.98430 

.19360 

.9.3107 

.2107(5 

.97754 

.22778 

.97:371 

.21174 

.0,60.501  .50 

11 

.17GS0 

1.98125 

.19395 

.08101 

.2iiai 

.97748 

. 2-2807 

.97.305 

.24.503 

.009.52!  40 

12 

.17708 

.93120 

.19423 

.930.)3 

.21132 

.97742 

.22835 

.97:35s! 

.21.531 

.!Mi01.5|  IS 

13 

.17737 

.98111 

.194.52 

.98.:90 

.21101 

.977:3.5 

.22.30:3 

.97:351  1 

.24.5.5!) 

.0(l!):i7  47 

14 

. 1 7703 

.98100 

.19481 

.93034 

.21189 

.97729 

.22892 

.97:3451 

.24587 

.000.30 1 40 

15 

.17791 

.98401 

i .19.509 

.98070 

.21218 

.97723 

.22920 

.97:338 

.21015 

.0!i0».3|  45 

IG 

.17811 

.9830.) 

! .19.5.38 

.98073 

.21240 

.97717 

.22948 

.973:31 

' .24C44 

.000101  44 

17 

.17852 

.98391 

.19500 

.9.8017 

.21275 

.97711 

.22977 

.9732.5 

.21072 

.OOOiX)!  4.3 

18 

.17880 

.98380! 

.19595 

.98031 

.21.303 

.97705 

.23005 

.97318 

.21700 

.000v)2j  42 

10 

.17900 

.98381! 

.19023 

.9'8).53 

.21331 

.97098 

.2:30:3:3 

.97.311 

.24728 

.008041  41 

20 

.1793 7 

.98378 

.19052 

.9.3050 

.21300 

.97092 

.23062 

.97304 

.247.50 

.00887!  40 

21 

.17900 

.93373 

.19080 

.93044 

.21.388 

.97aso* 

.2.3090 

.97208 

.21784 

. 90880 ' 39 

22 

.179.15 

.93308 

.19700 

.98030 

.21417 

.97080 

.2.3118 

.97-291 

.24813 

.9087,5:  88 

23 

.18021 1 

[.98302 

i .19737 

.9  5033 

.21415 

.97073 

.2,3140 

.97284 

.24841 

.90,3001  ‘37 

24 

.18052' 

.93357 

1 .19700 

.93027 

.21474 

.97007 

.2.3175 

.97278 

.24800 

.90.8.58:  30 

25 

.18081 

.93.352 

I .19791 

.93021 

.21502 

.97061 

.2.3-20.3 

.97271 

.24897 

.90851  ‘ ,35 

23 

.18100 

.93347 

! .19823 

.93013 

.21.530 

.97655 

.2.3231 

.97'204 

.24025 

.90844  31 

27 

.18138 

.98.341 

.19351 

.93010 

.21559 

.97613 

.2.3200 

.97257 

.24954 

.90.337  3:3 

23 

.18100 

.9.8.330 

.19830 

.93001  ' 

.21.537 

.97312 

.23288 

.97251 

.24982 

.90829  32 

29 

.18195 

.93331 

.19908 

.97003  1 

.21010 

.97633 

.28316 

.97244 

.25010 

.90822,  31 

30 

.18224 

.98335 

.19937 

.97002 

.21641 

.97630 

.23345 

.97237 

.25038 

.90815;  30 

31 

.18252 

.98320 

.19905 

.970378 

.21672 

.97023 

.2a37.3 

.97230 

.25066 

.90807 j 29 

32 

.18281 

.93315 

.19994 

.97031 

.21701 

.97617 

.23401 

.97223 

1 .2.5094 

.9(3800;  23 

33 

.18300 

.93.310 

.20022 

.9707.5 

.21720 

.97611 

.2.3420 

.97217; 

1 .25122 

.907931  27 

34 

.18338 

.90.304 

.20051 

.97030 

.21753 

.97301 

.23458 

.97210! 

1 .25151 

.90780,  26 

35 

.18307 

.93290 

.20079 

.97033 

.21733 

.97.503 

.23486 

.97203! 

1 .25179 

. 90778  i 25 

36 

.18395 

.93204! 

! .21103 

.97053 

.21314 

.97502 

.23.514 

.97190 

! .25207 

.90771  24 

37 

.18121 

.93238: 

i .20130 

.97052 

.21843 

.97535 

.23542 

.97189 

i .25235 

.98764  : 2:3 

38 

.184521 

.9.L’33' 

! .20105 

.97013 

.21871 

.97570 

.23571 

.97182; 

j .2.5263 

.9C7-.50l22 

39 

.18431 ' 

.93277! 

! .20103 

.97040 

.21300 

.97573 

.23599 

.97170, 

.25291 

.96749!  21 

40 

.18509 

.98272! 

! .20220 

.97034^1 

.21923 

.9 7563 j 

.23627 

.97169 

.25320, 

.96742  20 

41 

.18538 

.932071, 

; .20250 

.97023' 

.21953 

.97560 [ 

.23656 

.97162' 

1 .2.5348 

.907341  19 

42 

.18507 

.932011 

1 .20270 

.97022  ' 

.21035 

.97553 

.23684 

.97155; 

.25:370' 

.907271  18 

43 

.18595 

.93250! 

.2J307 

.97013  ; 

.2201-3 

.97547 

.23712 

.97148 

.25404 1 

.96719  17 

44 

.18624 

.93250 

.20330 

.97010  i 

.22041 

.975411 

.23740 

.97141! 

' .254:32! 

.90712!  10 

45 

.18052 ' 

.93215 

.20331 

.97.005 

.22070 

.97534 1 

.2.3769 

.97134 

; .25460 

.90705  15 

4G 

.18081 1 

.93240. 

.20303 

.97300 

.22003 

.97.523' 

.23797 

.97127 

1 .2.5488! 

.90097  14 

47 

.18710 

.9323L 

.20121 

.973.03 

.22126 

.97521; 

.2:3825 

.97120 

! .25516' 

.90090  13 

48 

.18738 

.93220, 

.20450 

.97337 

.22155 

.97.515: 

.2:3853 

.97113 

.255451 

.90082!  12 

49 

.18707 

.932231 

.20173 

.97331 

.221831 

.97503: 

.2.3882 

.97100 

.2.5.573! 

.9(3075  11 

50 

.187’95 

.98213, 

.20507 

.97375  ! 

.22212 

.97502 j 

.2:3910 

.97100, 

1 .2-5001 

. 90007 1 10 

51 

.18824 

.98212 

1 .20535 

.97869'! 

.22240 

.974961 

.23938 

.97093 

.25029 

. 96060 1 9 

52 

.18852 

.98207 

; .20503 

.97333,1 

.22268 

.97489 

.2:3966 

.970801 

1 .2.5657 

.900.>ii  8 

53 

.18881 

.98201 1 

! .20.502 

.978.5711 

.22207' 

.97483 

.2:3995 

.97079 

1 .25085 

.90015'  7 

54 

.18910 

.98190| 

! .203201.978514 

.223251 

.97476 

.24023 

.97072 

.2.5713 

.900.38:  0 

55 

.18938 

.981901 

.200401 

.97345  j 

.22.353 

.97470 

.24051 

.97065 

.2.5741 

.90630'  5 

5G 

.18907 

.98185! 

! .200771 

.978:30! 

.22.382, 

.97403! 

.24079 

. 97058 

.25769 

.90023!  4 

57 

.18995 

.981791 

I .20706' 

.97833' 

1 .22410; 

.97457; 

.24108 

.97051 

.2.5798 

.9(3015  3 

58 

.19024 

.981741 

1 .207.34 

.97827 

1 .224.38! 

.97450; 

.24130 

.97044 

.2.5820 

.90008  2 

59 

.19052  .981081 

i .207'03 

.97821! 

.22467 

.97444; 

.24104 

.970:37 

.2,58.54 

.90000  1 

GO 

.190811.98103! 

1 .20701 

.2‘2495 

.97437 

.24192 

.970:30 

.2.5882 

.90.593  0 

Cosin  1 Sine 

Cosin  ' 

Sine  1 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine  ; 

1 / 

79°  1 

i 78°  '! 

1 770 

76° 

75°  ! 

TABLES. 


651 


TABLE  VI. — Continued. 

Natural  Sines  and  Cosines. 


15“ 

16° 

17° 

18° 

1 19° 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

1 Sine 

Cosin 

T 

.25882 

.96593 

.27564 

.96126 

.29237 

.95030 

730902 

79510(5 

.32557 

114^52 

60 

1 

.25910 

.96585 

.27592 

.96118 

.29265 

.95622 

.30929 

.9.5097 

.32584 

.94542 

59 

2 

.25938 

.96578 

.27620 

.96110 

.29293 

.95613 

.30957 

.9.5088 

.32612 

.94533 

58 

3 

.2.5966 

.96570 

.27648 

.96102 

.29321 

.95605 

.30985 

.9.5079 

! .32639 

.94.523 

57 

4 

.25994 

.96562 

.27670 

.96094 

,29:348 

.95596 

.31012 

.95070 

.32067 

.94514 

56 

5 

.26022 

. 90555 

.27704 

.96086 

.29:376 

.95588 

.31040 

.95061 

.32694 

.94504 

55 

G 

.2G050 

.90547 

.27731 

.96078 

.29404 

.95579 

.31068 

.950.52 

.32722 

.94495 

54 

.2G079 

.90540 

.27759 

•96070 

.29432 

.95571 

.31095 

.95043 

.32749 

.94485 

53 

8 

.20107 

.90532 

.27787 

.96062 

.29460 

.95562 

.31123 

.9503.3 

.32777 

.94476 

52 

9 

.26135 

.90.524 

.27815 

.96054 

.29487 

.95554 

.31151 

.9.5024 

.32:04 

.94406 

51 

10 

.2G1G3 

.90517 

.27843 

.96040 

.29515 

.95545 

.31178 

.95015 

^ .32832 

.94457 

50 

11 

.2G191 

.90509 

.27871 

.90037 

.29.543 

.95536 

.31200 

.95000 

' .32859 

.94447 

4S 

12 

.2621 9 

.90502' 

.27899 

.96029 

.29571 

.95528 

.312.33 

.94997 

: .32887 

.94438 

48 

13 

.2G247 

.90194' 

.27927 

.98021 

.29599 

.95519 

.31201 

[.94988 

.32914 

.94428 

47 

14 

.20275 

.96480! 

.27955 

.90013 

.29626 

.95511 

.31280 

'.94979 

.32942 

.94418 

46 

15 

.2G303 

.90479 

.27983 

.96005 

.29654 

.95502 

.31:310 

.94970 

.32909 

.94409 

45 

IG 

.20331 

.96471' 

.28011 

.95997 

.29682 

.95493 

.31344! 

.94901 

.32997 

.94399 

44 

17 

.26359 

.90403 

.28039 

.95989 

.29710 

.95485 

.31372 

.94952 

.33021 

.94390 

43 

18 

.26387 

. 90450 

.28067 

.95981 

.29737 

.95476 

.31399 

.94943 

.33051 

.94380 

42 

19 

.2G415 

.90448 

.28095 

.95972 

.29705 

.95407 

.31427 

.94933 

: .33079 

.94370 

41 

20 

.26443 

.90440 

.23123 

.95964 

.29793 

.95459 

.31454 

.94924 

: .33100 

.94301 

40 

21 

.2G471 

.90-4.33' 

.28150 

. 95950 

.29821 

.95450 

.31482 

.94915 

' .33134 

.94351 

39 

22 

.2G500 

.90425 

.28178 

.95948 

.29849 

.95441 

.31510 

.94900 

.£3101 

.91342 

38 

23 

.20528 

.90417 

.28200 

.95940 

.29876 

.95433 

.31537 

.94897 

i .33189 

.94332 

37 

24 

.20556 

.90410 

.23234 

.95931 

.29904 

.95424 

.31505 

.94888 

.33216 

.94322 

36 

25 

.20584 

.90402 

.28202 

.95923 

.29932 

.95415 

.31593 

.94878 

[ .33244 

.94313 

35 

2G 

.20612 

.90304 

.28290 

.95915 

.29900 

.95407 

.31020 

.94809 

.33271 

.94303 

34 

27 

.20040 

.90380 

.23318 

.95907 

.29987 

.95398 

.31648 

.94800 

1 .33298 

.94293 

33 

28 

.26003 

.90379 

.23340 

.95898 

.30015 

.95389 

.31675 

.94851 

j .33320 

.942^ 

32 

29 

.20096 

.90371 

.23374 

.95890 

.30043 

.95:380 

.31703 

.94842 

! .33353 

.94274 

31 

30 

.26724 

.90303 

.23402 

.95882 

.30071 

.95372 

.31730 

.94832 

j .33331 

.94204 

30 

31 

.20752 

.90355 

.23429 

.95874 

.30098 

.95363 

.31758 

1.94823 

’ .33408 

.94254 

29 

32 

.26780 

.90347 

.23457 

.95805 

.30120 

.95354 

.31780 

,.94814 

.3:3430 

.94245 

28 

33 

.26808 

.90340 

.28485 

.95857 

.30154 

.95345 

.31813 

.94805 

i .33403 

.94235 

27 

34 

.26836 

.90332 

.23513 

.95849 

.30182 

.95337 

.31841 

.94795 

1 .33490 

.94225 

26 

35 

.26864 

.96324 

.23541 

.95841 

.30209 

.95328 

.31808 

.94780 

-33518 

.94215 

25 

36 

.26892 

.90310 

.28569 

.95832 

.30237 

.95319 

.31896 

.94777 

.3tiL)4o 

.94206 

24 

37 

,20920 

.90303 

.28597 

.95824 

.30205 

.95310 

.31923 

.94708 

' .33573 

.94196 

23 

38 

.26948 

.96301 

.28625 

.95810 

.30292 

.95301 

.31951 

.917'58 

1 .33000 

.94186 

22 

39 

.20976 

.90203 

.28652 

.95807 

.30320 

.95293 

.31979 

.94749 

.33027 

.94176 

21 

40 

.27004 

.90285 

.28080 

.95799 

.30348 

.95284 

.32006 

.94740 

.33055 

.94167 

20 

41 

.27032 

.9027?' 

.28708 

.95791 

.30376 

.95275 

.32034 

.94730 

.33082 

.94157 

19 

42 

.27000 

.90200 

.28730 

.95782 

.30403 

.95266 

.32001 

.94721 

.33710 

.94147 

18 

43 

.27088 

.90201 

.28704 

.95774 

.30431 

.95257 

.32089 

.94712 

.33737 

.941:37 

17 

44 

.27116 

.90253, 

.28792 

.9.5706 

.30459 

.95248 

.32116 

.94702 

.33764 

.94127 

16 

45 

.27114 

.90246; 

.28820 

.95757 

.30486 

.95240 

.32144 

.94093 

.33792 

.94118 

15 

4G 

.27172 

.902.38' 

.28847 

.9.5749 

.30514 

.95231 

.32171 

.94084 

.33819 

.94108 

14 

47 

.27200 

.962.30 

.28875 

.95740 

.305421 

.95222 

.32199 

.94074 

.33846 

.94098 

13 

48 

.27228 

90222 

.28903 

.9.57.32 

.30570! 

.95213! 

.32227 

.94665 

.33874 

.94088 

12 

49 

.27250 

; 90214 

.28931 

[.9.5724 

.30.597 

.95204! 

.32254 

.94050 

33901 

.94078 

11 

50 

.27284 

90200 

.28959 

.95715 

.30025 

.95195 

.32282 

.94046 

1 .33929 

.94008 

10 

51 

.27312 

'.90198 

.2898?! 

.9.5707 

.30653 

.95186 

.32309 

.94637 

1 .a3956 

.94058 

9 

52 

.27340 

.90190 

.29015: 

.95698 

.30080 

.95177 

.32337 

.94027 

i .33983 

.91019 

8 

53 

.27368 

.90182 

.29042 

.9.5090 

.30708 

.95108 

.32304 

.94018 

.34011 

.91039 

7 

54 

.27396 

.96174 

.29070 

.9.5681 

.30730 

.951.59 

.32392 

.94009 

1 .34038 

.94029 

6 

55 

.27424 

.90I0ii 

.29098 

.9.5073 

.30703 

.951.50 

.32410 

.94599 

1 .34005 

.94019 

5 

5G 

.27452 

.901.58 

.29126 

.95004 

i .30791 

.95142 

.32447 

.94.590 

1 .34093 

.94009 

4 

57 

.27480 

.90150 

.291.54 

.9.50.50 

.30819 

.951:3:3 

.32474 

.93580 

i .34120 

.93999 

3 

58 

.27508 

.90142 

.20182 

.9.5047 

.30846 

.95124 

.32.502 

.94571 

1 .34147 

.93989 

2 

59 

.27.536 

.061.34 

.20209 

.9.50.39 

.30874 

.95115 

.32529 

.94501 

! .341751 

.93979 

1 

GO 

27.504 

.9(5126 

.292.37: 

.9.5030 

.30902 

.95106 

.32.5.57 

.94552 

1 .34202| 

.93909 

0 

/ 

L’osin 

Sine  1 

Cosin  1 

Sine 

Cosin 

Sine  1 

Cosin 

Sine 

j Cosin  1 

Sine 

/ 

740  1 

73° 

1 72°  1 

71° 

0 

0 

452 


TABLE  VI.  — Contiuued. 

Natural  Sines  and  Cosines. 


20“ 

21»  1 

22" 

23" 11 

e 

Sine 

Cosin 

Sino 

Cosin 

Sino 

Cosin 

Sine 

Cosin 

Sin*» 

C’osin  1 

0 

.34202 

.93909 

.:4.5a37 

.9a‘4.58 

.:4740i 

.9*2718 

.39073 

.9*20.5) 

I .40074 

.9i:i55' 

00 

1 

.34229 

.939.59 

..‘4.")8(;4 

.9:4:448 

.374H8 

. 9*2707 1 

..39I(X) 

.920:49 

1 .40700 

.91:313 

59 

2 

..342.57 

.9:4949 

..3.5891 

.9:4:i‘i7 

.37515 

.9*2097 

..‘491*27 

.920*28 

1 .40727 

.91:331 

58 

3 

..312,S4 

.9:49.39 

.:3.5918 

.9:4:427! 

..‘47512 

.9*20801 

..‘491.54 

.9*2010 

, .407.53 

.01319 

57 

4 

..34311 

.93929 

.:3594.5 

.9:4:410' 

..37.509 

.9*2075 

..‘49180 

.92005 

' .40780 

.91.307 

.V, 

5 

.343.39 

.9:4919 

.:3.5973 

.9.a‘406 

.37.595 

.9*2004 

..39*207 

.91991 

1 .40800 

.91295 

55 

G 

.34.300 

.9:4909 

..3(5000 

.9:4295 

.37022 

.920.53 

' ..392‘M 

.91982 

' .40833 

.91253 

.51 

7 

..‘34.393 

.9.3899 

.:3(‘>()27 

.9:42R5 

.370-19 

.9*20-12', 

.;49*200 

.91971 

! .40800 

.91 ‘272  i 

.53 

8 

..‘34121 

.9:4889; 

..‘50().->4 

. 9:4274  > 

.37070 

.9*20:41 ' 

j ..39*287 

.919.59 

1 .40880 

.91200; 

52 

9 

..‘344  48 

.9:4879 

.:50081 

.9:4204 

.3770:4 

.9*2020 

! ..39:111, 

.91918 

; .40913 

.91 218 1 

51 

10 

.34475 

.93809 

.:30108 

.9.4253 

.377:40 

.92009 

■ ..39311 

.919.30 1 

.409.39 

.912361 

50 

11 

..34.503 

.93a59 

..30ia5 

.9.3213 

.377.57 

.92.598 

.39307 

.91925 

! .409001 

.91221! 

49 

12 

..‘345.‘30 

.9:4,849 

..‘40102 

.9:4*2:42 

..37781 

.9*2587 

, .39:494' 

.91911 

1 .409!)2| 

.91212 

48 

13 

.34,5.57 

.9:4839 

1 .30190 

.93222 

.37811 

.9*2,570 

.:494  21, 

.91902 

.41019| 

.91200 

47 

14 

.34.584 

.9:4829 

1 .:30217 

.9:4211 

.37848 

.92505' 

..39418; 

.91891 

, .410451 

.91188 

40 

15 

.34012 

.9.3819 

I ..3024  4 

.9:4201 

..3780.5 

.92554; 

.39474 

.91879, 

.41072! 

.91170 

10 

..340.‘59 

.93809 

! .30271 

.93190 

.37892 

.92543' 

' .;49.501 

.91808 

, .41098; 

.911(34 

! 41 

17 

.34000 

.9:4799 

1 .30298 

.93180, 

..37919 

.92532 

..39528 

.91K50 

i .41125, 

.91152 

43 

18 

.31094 

.93789 

; ..30.325 

.93109 

.37940 

.92521 ' 

..395.55 

.91M.5 

j .411.511 

.91140 

42 

19 

.34721 

.93779 

i .30.3.52 

.93159 

.37973 

. 9*2510  i 

..‘49.581 

'.918.53 

.41178; 

,.91128 

41 

20 

.34748 

.93709 

1 .36379 

.93148 

.37999 

.92499, 

.39008 

.9182*2 

.41204 

.91110 

40 

21 

.34775 

.937.59 

! .36406 

.931.37 

..38020 

.92188 

.390.a5 

.91810 

.41231 

.91101 

39 

22 

.31803 

.93748 

1 ..304:34 

.93127 

.380.5.3 

.924771 

.:49001 

.91799 

; .41257 

1.91092 

38 

23 

.34830 

.9:47:48 

.30401 

.9:4116 

.38080 

.92400 

.39688 

.91787 

.41284 

'.91080 

37 

24 

.34857 

.9.3728 

! .30488 

! .9.3100 

.38107 

.92455 

.39715 

.91775 

.41.310 

.91008 

30 

25 

.34834 

.93718 

1 .30515 

.9.3095 

.381.34 

.9244-41 

.39741 

.91704 

! .41.537 

.91056 

35 

26 

.34912 

.93708 

..30,542 

' .9.3084 

1 .38101 

.924321 

..39768 

.91752 

; .41.303 

.91044 

34 

27 

.34939 

.93098 

.30509 

.93074 

.38188 

.92421 

.39795 

.91741 

.41.390 

.910.32 

33 

28 

.34966 

.9.3688 

.30590 

.9.3003 

.38215 

,.92410 

.39822 

.91729 

1 .41416 

.91020 

32 

29 

.34993 

.93077 

.30023 

.9.3052 

.38241 

,.92399 

.39848 

.91718 

.41443 

.91008 

31 

30 

.35021 

.93007 

,30650 

.93042 

.38268 

,.92388 

.39875 

.91700 

.41409 

.90996 

130 

31 

.35048 

.9.36.57 

.36677 

.9.3031 

.38295 

!.  92377 

.39902 

.91694 

.41496 

.90984 

' 29 

32 

.35075 

.9:4047 

.367(44 

.93020 

.38322 

; .92300 

.39928 

.91083 

.41522 

.9097-2 

28 

33 

.35102 

.9.3037 

.36731 

.9:4010 

i .38819 

i .92355 

' .39955 

.91671 

.41549 

.90960 

27 

34 

.351.30 

.9:3026 

.36758 

.92999 

.38376 

;.  92343 

.39982 

.91600 

,41575 

.90948 

20 

35 

.35157 

.9:3616 

.36785 

.92988 

.38403 

.92232 

.40008 

.91648 

.41602 

.90936 

25 

30 

.35184 

.9.3606 

.36812 

.92978 

.38430 

.92321 

.400.35 

.91636 

.41628 

.90924 

24 

37 

.35211 

.93596 

.36839 

.92907 

.38456 

.92310 

.40002 

.91625 

.416.55 

.90911 

23 

38 

.35239 

. 93585 

.36807 

.92950 

.38483 

.92299 

.40088 

.91613 

.41681 

.90899 

22 

39 

.35266 

. 9:3575 

.36894 

.92945 

.38510 

.92287 

.40115 

.91601 

; .41707 

.90887 

1 21 

40 

.35293 

. 93505 

.36921 

.92935 

.38537 

.92276 

.40141 

.91590 

1 .41734 

.90875 

20 

41 

.35320 

.93555' 

.36948 

.92924' 

.38564 

.92265' 

.40168 

.91578 

.41760 

.90863 

! 19 

42 

.35347 

.93544 

.36975 

.92913 

.38591 

.92254; 

.40195 

.91566 

1 .41787 

.90851 

18 

43 

.35375 

.93534 

.37002 

.92902 

.38017 

.92243 

.40221 

.91555 

1 .41813 

.90839 

17 

44 

.35402 

.93524 

.37029 

.92892 

.38644 

.922311 

.40248 

.91543 

! .41840 

.90826 

16 

45 

.35429 

.93514 

.37056 

.92881 

.38671 

.922*20 

.40275 

.91531 

! .41866 

.90814 

15 

46 

.35456 

.93503 

.37083 

.92870 

.38698 

. 92209 1 

.40301 

.91519 

1 .41892 

.90802 

14 

47 

.35484 

.93493 

.37110 

.92859 

.38725 

.92198 

.40328 

.91.508 

j .41919 

.90790 

13 

48 

.355111 

.93483 

.371.37 

.92849 

.38752 

.92186 

.40355 

.91496 

.41945 

.90778 

12 

49 

.35538 

.9:3472 

. .37104 

.92838 

.38778 

.92175 

.40381 

.91484 

.41972 

1.90766 

11 

50 

.35505 

.93462 

.37191 

.92827 

.38805 

.92164 

.40408 

.91472 

1 .41998 

.90753 

10 

51 

.35.592 

.9.3452 

.37218 

.92816 

.38832 

.92152 

.40434 

.91461 

.42024 

.90741 

9 

52 

.a5619 

.9:3441 

.37245 

.92805 

.38859 

.92141 

.40401 

.91449 

I .42051 

.90729 

8 

53 

.35647 

.9:3431 

.37272 

.92794 

.38886 

.92130 

.40488 

.91437 

j .42077 

.90717 

7 

54 

.35674 

.9:3420 

.37299 

.92784 

.38912 

.92119 

.40514 

.91425 

.42104 

.90704 

6 

55 

.3.5701 

.9:4410 

.:37.326 

.92773 

.38939 

.92107 

.40541 

.91414 

.42130 

.90692 

5 

56 

..35728 

.93400 

.37353 

.92762 

.38966 

.92096 

.40567 

.91402 

.42156 

.90680 

4 

57 

..3.57.55 

.93:389 

..37.380 

.92751 

.38993 

.92085 

.40594 

.91390 

.42183 

.90668 

3 

68 

..35782 

.9.3:379 

.37407 

.92740 

,39020 

.92073 

.40621 

.91378 

.42209 

.90655 

2 

59 

..3.5810 

.93:308 

.37434 

.92729 

.39046 

.92062 

.40647 

.91366 

.42235 

.90643 

1 

60 

..35837 

.93:358 

.:47401 

.92718 

.39073 

.920.50 

.40074 

.913.55 

.42262 

.90631 

0 

/ 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

/ 

69» 

68" 

67" 

66° 

65" 

TABLES. 


653 


TABLE  VI. — Coniinued. 

Natural  Sines  and  Cosines. 


25° 

26“ 

27“ 

I 

00 

29° 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin  1 

Sine 

Cosin 

0 

.42262 

.90631 

.43837 

.89879 

.45399 

.89101 

.46947 

.88295 

.48481 

.87462 

60 

1 

.42288 

.90618 

.43863 

.89867 

.45425 

.89087 

.46973 

.88281 

.48506 

.87448 

59 

2 

.42315 

.90606 

.43889 

.89854 

.45451 

.89074 

.46999 

.88267 

.48532 

.87434 

58 

3 

.42341 

.90594 

.43916 

.89841 

.45477 

.89061 

.47024 

.88254 

.48557 

.87420 

57 

4 

.42367 

.90582 

.43942 

.89828 

.45503 

.89048 

.47050 

.88240 

.48583 

.87406 

56^ 

5 

.42394 

.90.569 

.43968 

.89816 

.45529 

.89035 

.47076 

.88226 

.48608 

.87391 

55 

6 

.42420 

.90557 

.43994 

.89803 

.45554 

.89021 

.47101 

.88213 

.48634 

.87377 

54 

7 

.42446 

.90545 

.44020 

.89790 

.45580 

.89008 

.47127 

.88199 

.48659 

.87363 

53 

8 

.42473 

.905.32 

.44046 

.89777 

.45606 

.88995 

.47153 

.88185 

.48684 

.87349 

52 

9 

.42499 

.90520 

.44072 

.89764 

.45632 

.88981 

.47178 

.88172 

.48710 

.87335 

51 

10 

.42525 

.90507 

.44098 

.89752 

.45658 

.88968 

.47204 

.88158 

.48735 

.87321 

50 

11 

.42552 

.90495 

.44124 

.89739 

.45684 

.88955 

.47229 

.88144 

.48761 

.87306 

49 

12 

.42578 

.90483 

.44151 

.89728 

.45710 

.88942 

.47255 

.88130 

.48786 

.87292 

48 

13 

.42604 

.90470 

.44177 

.89713 

.45736 

.88928 

.47281 

.88117 

.48811 

.87278: 

47 

14 

.42631 

.90458 

.44203 

.89700 

.45762 

.88915 

.47306 

.88103 

.48837 

.87264 

46 

15 

.42657 

.90446 

.44229 

.89687 

.45787 

.88902 

.47332 

.88089 

.48862 

.87250 

45 

16 

.42683 

.90433 

.44255 

.89674 

.45813 

.88888 

.47358 

.88075 

.48888 

.87235 

44 

17 

.42709 

.90421 

.44281 

.89662 

.45839 

.88875 

.47383 

.88062 

.48913 

.87221 

43 

18 

.42736 

.90408 

.44307 

.89649 

.45865 

.88862 

.47409 

.88048 

.48938 

.87.207 

42 

19 

.42762 

.90.390 

.44333 

.896.36 

.45891 

.88848 

.47434 

.8803-1 

.48964 

.87193 

41 

20 

.42788 

.90383 

.44359 

.89623 

.45917 

.88835 

.47460 

.88020 

.48989 

.87178 

40 

21 

.42815 

.90371 

.44385 

.89610 

.45942 

.88822 

.47486 

.88006 

.49014 

.87164 

39 

22 

.42841 

.90358 

.44411 

.89597 

.45908 

.88803 

.47511 

.87993 

.49040 

.87150 

38 

23 

.42867 

.90346 

.44437 

.89584 

.45994 

.88795 

.47537 

.87979 

.49065 

.87136 

37 

24 

.42894 

.903:14 

.44464 

.89571 i 

.40020 

.88782 

.47502 

.87905 

.49090 

.87121 

36 

25 

.42920 

.90321 

.44490 

.89558! 

.40046 

.88708 

.47588 

.87951 

.49116 

.87107 

35 

26 

.42946 

.90309 

.44516 

.89545 

.40072 

.88755 

.47614 

.87937 

.49141 

.87093 

34 

27 

.42972 

.90296 

.44542 

.89532 

.40097 

.88741 

.47039 

.87923 

.49166 

.87079 

33 

28 

.42999 

.90234 

.44568 

.89519 

.46123 

.88723 

.47665 

.87909 

.49192 

.87064 

32 

29 

.43025 

.99271 

.44594 

.89506 

.40149 

.88715 

.47690 

.87896 

.49217 

.87050 

31 

30 

.43051 

.90259 

.44620 

.89493 

.46175 

.88701 

.47716 

.87882 

.49242 

.87036 

30 

31 

.43077 

.90246 

.44646 

.89480 

.46201 

.88688 

.47741 

.87868 

.49268 

.87021 

29 

32 

.43104 

.90233 

.44672 

.89467 

.40220 

.88074 

.47767 

.87854 

.49293 

.87007 

28 

33 

.43130 

.90221 

.44698 

.89454 

.46252 

.88661 

.47793 

.87840 

.49318 

.86993 

27 

34 

.43156 

.90208 

.44724 

.89441 

.46278 

.88647 

.47818 

.87826 

.49344 

.86978 

26 

35 

.43182 

.90196 

.44750 

.89428 

.46304 

.8863-1 

.47844 

.87812 

.49369 

.86964 

25 

36 

.43209 

.90183 

.44776 

.89415 

.40330 

.88620 

.47869 

.87798 

.49394 

.86949 

24 

37 

.43235 

.90171 

.44802 

.89402 

.46355 

.88607 

.47895 

.87784 

.49419 

.86935 

23 

38 

.43261 

.901.58 

.44828 

.89389 

.46381 

.88593 

.47920 

.87770 

.49445 

.86921 

22 

39 

.43287 

.90146 

.44854 

.80376 

.40407 

.88580 

.47946 

.87756 

.49470 

.86906 

21 

40 

.43313 

.90133 

.44880 

.89363 

.40433 

.88566 

.47971 

.87743 

.49495 

.86892 

20 

41 

.43340 

.90120 

.44906 

.89350 

.46458 

.88553 

.47997 

.87729 

.49521 

.86878 

19 

42 

.43366 

.90103 

.44932 

.89337 

.46484 

.88539 

.48022 

.87715 

.49546 

.86863 

18 

43 

.43392 

.90095 

.44958 

.89324 

.46510 

.88526 

.48048 

.87701 

.49571 

.86849 

17 

44 

.43418 

.90082 

.44984 

.89311 

.46536 

.88512 

.48073 

.87687 

.49596 

.86834 

16 

45 

.43445 

.90070 

.45010 

.89298 

.46561 

.88499 

.48099 

.87673 

.49622 

.86820 

15 

46 

.43471 

.90057 

.45036 

.89285 

.46587 

.88485 

.48124 

.87659 

.49647 

.86805 

14 

47 

.43497 

.90045 

.45062 

.89272 

.46613 

.88472 

.48150 

.87045 

.49672 

.86791 

13 

48 

.43523 

.90032 

.45088 

.89259 

.46639 

.88458 

.48175 

.87031 

.49697 

.86777 

12 

49 

.43549 

.90019 

.45114 

.89245 

.40064 

.88445 

.48201 

.87617 

.49723 

.86762 

11 

50 

.43575 

.90007 

.45140 

.89232 

.40690 

.88431 

.48226 

.87603 

.49748 

.86748 

10 

51 

.43602 

.89994 

.45166 

.89219 

.46716 

.88417 

.48252 

.87589 

.49773 

.86733 

9 

52 

.43628 

.89931 

.45192 

.89206 

.46742 

.88404 

.48277 

.87575 

.49798 

.86719 

8 

53 

.43654 

.89968 

.45218 

.89193 

.46767 

.88390 

.48:303 

.87501 

.49824 

.86704 

7 

54 

.43680 

.89956 

.45243 

.89180 

.46793 

.88377 

.48328 

.87546 

.49849 

.86690 

6 

55 

.43706 

.89943 

.45269 

.89167 

.46819 

.88363 

.48354 

.87532 

.49874 

.86675 

5 

66 

.43733 

.89930 

.45295 

.89153 

.46844 

.88349 

.48379 

.87518 

.49899 

.86661 

4 

67 

.43759 

.89918 

.45321 

.89140 

.46870 

.88336 

.48405 

.87504 

.49924 

.86646 

3 

58 

.43785 

.89905 

.45347 

.89127 

.46896 

.88322 

.484:30 

.87490 

.49950 

.86632 

2 

59 

.4:1811 

.89892 

.4.5373 

.89114 

.46921 

.88:308 

.48456 

.87476 

.49975 

.86617 

1 

60 

.43837' 

.89879 

.45:599 

.89101 

.46947 

.88295 

.48481 

.87462 

.50000 

.86603 

_0 

$ 

Cosin  1 Sine 

Cosin 

Sine 

1 Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine  ! 

/ 

63“ 

1 62“ 

61“ 

60° 

654 


SURVEYING. 


TABLE  V\.—CvuiiutuE 

Natural  Sines  and  Cosines. 


CO 

o 

31“  i' 

|co 

0 

83°  II 

34^ 1 

Sine 

Cosin 

Sine 

Cosi  n 

Sine 

Cosin  ^ 

Sine  ' 

CosfnII 

Bine  ICosIn 

0 

..50000 

.86603 

.51.504 

.857171 

752992 

; 84805  j 

'.M424 

.83867,' 

.55919! 

.82904 

60 

1 

..50025 

.86.588 

.51.529 

.8.5702, 

.5;K)17 

.84789  : 

..5-1 IKH. 

.83851 , 

.6r)913' 

.82887 

.59 

2 

.50050 

.86573 

.51.5.54 

.8,5687 

.5.3041 

.84774 

..5-1.513 

.85835" 

..5.5!HWl 

,82.871 ' 

.58 

3 

..50076 

.86.5.59 

.51.579 

.8.5672:1 

..5.3066 

,847.59 

..54.5:37 

.83819 

..259921 

.82855 

.’•,7 

4 

..50101 

.86514 

.51604 

.8.56.57; 

.5:5091 

.81743 

.5-1.561 

.83804, 

.56016 

.828:39 

6 

..50126 

.86.5.30 

.51628 

.a5642 

.r).3115! 

.84728 

..54.')K6 

.83788 

..56(M0 

.82822 

.5.5 

6 

..50151 

.86.515 

.516.53 

.8.5627 

.53140 

.84712 

..5-1610 

.83772 

..'•>6061 

.82800 

54 

7 

..50176 

.86.501 

.51678 

.a5612 

,.531M 

.81697 

.,546.35 

.817.56 

..560H8 

.82790 

.53 

8 

.50201 

.86486 

.51703 

.85597 

.!>31H9 

.84681 

..516.59 

.83710; 

..56112 

.827'7:3i 

f -J 

9 

..50227 

.86171 

.51728 

.R5.5S2 

..53214 

.84066 

.51(583 

.83724  , 

..561.36 

.82757 

5. 

10 

.50252 

.86457 

.51753 

.85567 i 

.53238 

.84650 

.54708 

.837081 

, .56160 

.827411 

50 

11 

.50277 

.80412 

.51778 

.8.5.5.51 ' 

..5.3263 

.846.35' 

..54732' 

.83692' 

..56184 

.82724' 

49 

12 

.50302 

.86427 

.51803 

.a5.5.36| 

.. 5:4288 

.81619 

..517.50 

.85676 

..56208 

.82708 

48 

13 

..50327 

.86113 

.5182.8 

.a5.521  1 

..5.'5312! 

.81001 

..51781  ' 

.83(360 

.5(7'32 

.82692 

47 

14 

..503.52 

.86.391 

.518.52 

.a5.506 

.5.33.37 

.81.588 

..54^05' 

.83615 

..':-32.56 

.82675 

46 

15 

.50377 

.86.384 

.51877 

.85491 

.,5:4.361 

.84;>73 

.548291 

.83029 

.56280 

.826.59 

45 

10 

.50403 

.86.369 

.51902 

.a5470 

.5.2386 

,81.5.57 

.51854' 

.83(313 

.,56.305 

.82013 

44 

17 

..50428 

.86.354 

.51927 

.85101 1 

..53411 

.81.542 

..51878 

. 83.597 1 

; .56.329 

.82026 

43 

18 

..504.53 

.86.340 

.519.52 

.8.54-16 

.5:44.35 

.84.526 

.54902 

.82581 1 

! ..56:3.53 

.82610 

42 

19 

.50478 

.86325 

.51977 

.a5431 

..53-160 

.84511 

..'.4927 

.8:3.5651 

.56.377 

.82.593 

41 

20 

.50503 

.80310 

.52002 

.85416 

.58484 

.84495 

.54951 

.83249 

.56401 

.82577, 

40 

21 

..50528 

.86295 

..52020 

.a5401 

.53.509 

.84480 

..54975 

.8.3523 

.56425 

.8256r 

39 

22 

.50.553 

.80281 

..5,20.51 

.85335 

.5.3534 

.81404 

i ..54999 

.83517 

.56449 

.82544 

.38 

23 

.50578 

.802.50 

. .520  ( 6 ' 

.^370 

.53558 

.81418 

j .5.5024 

.82501 

..50473 

.82528 

37 

24 

..50603 

.862.51 

.52101 

. 85.3.55 ! 

.5:4583 

.814:23 

1 .5.5048 

.83485 

.56497 

.82.511 

36 

25 

.50628 

.802.37 

1 .52126 

.a5340. 

.5.3007 

.81417 

1 .55072 

.82169 

.56.521 

.82-495 

35 

26 

.500.54 

.86222 

..52151 

.8.5325. 

.5.3032 

.81402 

..5.5097 

.84453 

.56.545 

.82478 

34 

27 

..50679 

.86207 

..52175 

.a53l0 

..5.3056 

.84280 

.55121 

.834.37 

.50.569 

.82462 

33 

28 

.50704 

.80192 

..52200 

.a5294 

.53681 

.81:570 

.55145 

.83421 

.56.593 

.8:^6 

32 

29 

.50729 

.861731 

..52225 

.a5279 

.53705 

.813.55 

.5.5109 

.82405 

.56617 

.82-129 

31 

30 

.50754 

.80103 

.52250 

.85204. 

.53730 

.84339 

.55194 

. 83389 1 

.56641 

.8;«i:3 

30 

31 

.50779 

.80148' 

.52275 

.85249 

..53754 

.84.324 

.55218 

.82373 

.56665 

.82.396 

' 29 

32 

.50804 

.801331 

.52209 

.a5234 

.53779 

.84303 

.55242 

.83356 

.56089 

.82:380 

28 

33 

.50829 

.80119 

.52324 

.85218 

.53804 

.84292 

.t52CG 

.83340 

.56713 

.82363 

27 

U 

.50854 

.8010L 

.52349 

.85203 

.53828 

.84277 

.55291 

.83.3241 

.56730 

.82347 

26 

35 

.50879 

.80089 

.52374 

.85188 

.53853 

.84261} 

.55315 

.82308 

.56760 

.82.3.30 

25 

36 

.50904 

.80074, 

.52399 

.85173 

.53877 

.84245 

.55239 

.8.3292 

.56784 

.82314 

24 

37 

.50929 

.80059' 

.52423 

.85157 

.53902 

.84230 

.55.363 

1.83276 

.56808 

.82297 

23 

38 

.50954 

.86045 j 

.52-443 

.85142 

.53926 

.84214 

.55388 

1.8.3260 

.56832 

.82281 

22 

39 

.50979 

.86030; 

.52173 

.85127 

.53951 

.84198 

.55412 

i .8:3244, 

.56856 

.82264 

21 

40 

.51004 

.86015} 

.52493 

.85112 

.53975 

.84182 

.55436 

.83228 

.50880 

.82248 

20 

41 

.51029 

.86000 

.52522 

.85096 

.54000 

.84167 

,55460 

.832121 

.56904 

.82231 

' 19 

42 

.51054 

.85985 

.52547 

.85031 

.54024 

.84151 

.55484 

.83195! 

.56928 

.82214 

18 

43 

.51079 

.85970 

.52572 

.85066 

.54049 

.84135 

.55509 

.831791 

.56952 

82198 

. 17 

44 

.51104 

.85956 

.52597 

.85051 

.54073 

.84120 

.55533 

.831631 

.56970 

.82181 

1 16 

45 

..51129 

.85941 

.52021 

.85035 

.54097 

.84104 

.55557 

.831471 

.57000 

.82165 

15 

46 

.51154 

.85926 

.52646 

.85020 

.54122 

.84088 

.55581 

.831.31 1 

.57024 

.82148 

14 

47 

.51179 

.85911 

.52071 

.85005 1 

.54146 

.84072 

.55605 

.83115: 

.57047 

.82132 

13 

48 

.51204 

.8.5896 

.52096 

.84989 

.54171 

.84057 

.55630 

.83098; 

.57071 

.82115 

12 

49 

.51229 

.85881 

.52720 

.84974 

.54195 

.84041 

.55654 

.83082, 

.57095 

.82098 

; 11 

50 

.51254 

.85860 

.52745 

.84959 

.54220 

.84025 

. 55678 

.83066, 

.57119 

.82082 

10 

51 

..51279 

.8.5851 

.52770 

.84943 

.54244 

.84009 

.55702 

.830501 

.57143 

.82065 

9 

52 

.51304 

.85330 

.52794 

.84928 

,54269 

.83994 

.55726 

.83034: 

.57167 

.82048 

8 

53 

.51329 

.85821 

.52819 

.84913 

.54293 

.83978 

.55750 

.83017; 

.57191 

.82032 

7 

54 

.51.354 

.85800 

.52844 

.84897 

.54317 

.83962 

.55775 

.83001 1 

.57215 

.82015 

6 

55 

.51379 

.85792 

.52869 

.84882 

.54.342 

.8.3946 

. 55799 

.82985, 

..57238 

.81999 

5 

56 

.51404 

.8.5777 

.52893 

.84866 

.54.366 

.83930 

.55823 

.829091 

.57262 

.81982 

4 

57 

.51429 

.8.5762 

.52918 

.84851 

.54.391 

.83915 

..55847 

.82953} 

.57286 

.81965 

3 

58 

.514.54 

.8.5747 

.52943 

.84836: 

.54415 

.83899 

.5.5871 

.829.36 

.57310 

.81949 

8 

59 

.51479 

.85732 

.52907 

.848201 

.54440 

.8:3883 

..55895 

.82920 

.573.34 

.819:32 

1 

.51.504 

.85717 

..52992 

.84805 

.54404 

.83867 

.5.5919 

.82904 

.573.58 

.81915 

1 J 

/ 

Cosin 

Sine 

Cosin 

1 Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

1 f 

69° 

68° 

67° 

66“ 

65“  1 

1 

TABLES. 


655 


TABLE  VI. — Continued. 

Natural  Sines  and  Cosines. 


35° 

36° 

37° 

0 

CO 

CO 

39° 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

/ 

0 

.57358 

.81915 

.58779 

.80902 

.60182 

.7‘9864 

761506 

778801 

762932 

.77715 

1 

.57381 

.81899 

,58802 

.80885 

.60205 

.79846 

.61589 

.78783 

.62955 

.77696 

59 

2 

.57405 

.81882 

.58826 

.80867 

.60228 

.79829 

.01012 

.78705 

.62977 

1.77678 

58 

3 

.57429 

.81865 

.58849 

.80850 

.60251 

.79811 

.61635 

.78747 

.63000 

.77660 

57 

4 

.57453 

.81848 

.58873 

.80833 

.60274 

.79793 

.61658 

.78729 

.63022 

.77'641 

56 

5 

.57477 

.81832 

.58896 

.80816 

.60298 

.79776 

.61081 

.78711 

.63045 

1.77623 

55 

6 

.57501 

.81815 

.58920 

.80799 

.60321 

.79758 

.01704 

.78694 

.63068 

1.77605 

54 

7 

.57524 

.81798 

.58943 

.80782 

.60344 

.79741 

.61726 

.78676 

.63090 

1.77586 

53 

8 

.57548 

.81782 

.58967 

.80705 

.60367 

.79723 

.61749 

.78658 

.63113 

.77568 

52 

9 

.57572 

.81765 

.58990 

.80748 

.60390 

.79706 

.61772 

.78640 

.63135 

.77550 

51 

10 

.57596 

.81748 

.59014 

.80730 

.60414 

.79688 

.61795 

.78622 

.63158 

.77531 

50 

11 

.57619 

.81731 

.59037 

.80713 

.60437 

.79671 

.61818 

.78604 

.63180 

.77513 

40 

12 

.57643 

.81714 

.59061 

.80096 

.60460 

.79653 

.61841 

.78586 

.03203 

.77494 

48 

13 

.57067 

.81698 

.59084 

.80679! 

.60483 

.79035 

.61864 

.78508 

.03225 

.77476 

47 

14 

.57091 

.81081 

.59108 

.80002 

.60506 

.79018 

.61887 

.78550 

.03248 

.77458 

46 

15 

.57715 

.81004 

.59131 

.80644 

.60529 

.79000 

.61909 

.78532 

.63271 

i. 77439 

45 

16 

.57738 

.81047 

.59154 

.80627 ! 

.60553 

.79.583 

.61932 

.785141 

.63293 

.77421 

44 

17 

.57762 

.81631 

.59178 

.80610, 

.60576 

.79505 

.61955 

.78496; 

.03310 

.77402 

43 

18 

.57786 

.81014 

.59201 

.80593 

.60599 

.79547 

.61978 

.78478 

.03338 

.77384 

42 

19 

.57810 

.81597 

.59225 

.80576 

.60622 

.79530 

.62001 

.78460 

.63361 

.77366 

41 

20 

.57833 

.81580 

.59248 

.80558 1 

.60645 

.79512 

.62024 

.78442 

.03383 

.77347 

40 

21 

.57857 

.81563 

.59272 

.805411 

.60668 

.79494 

.62046 

.78424 

.03406 

.77329 

39 

22 

.57881 

.81546 

.59295 

.80524 

.60091 

.79477 

.62069 

.78405 

.63428 

.77310 

38 

23 

.57904 

.81530 

.59318 

.80507 

.60714 

.79459 

.62092 

.78387 

.63451 

.77292 

37 

24 

.57928 

.81513 

.59342 

.80489 

.60738 

.79441 

.02115 

.78309 

.63473 

.77273 

36 

25 

.57952 

.81496 

.59305 

.80472 

.60761 

.79424 

.62138 

.78351 

.63496 

.77255 

35 

23 

.57976 

.81479 

.59389 

.80455' 

.60784 

.79406 

.62160 

.78333 

.63518 

.77236 

34 

27 

.57999 

.81462 

.59412 

.80438 

.60807 

.79388 

.62183 

.78315 

.03540 

.77218 

33 

28 

.58023 

.81445 

.59436 

.80420 

.60830 

.79371 

.62206 

.78297 

.03563 

.77199 

32 

29 

.58047 

.81428 

.59459 

.80403 

60853 

.79353 

.62229 

.78279 

.03585 

.77181 

31 

SO 

.58070 

.81412 

.59482 

.80380, 

.60876 

.79335 

.62251 

.78201 

.03608 

.77162 

30 

31 

.58094 

.81395 

.59506 

.80368' 

.60899 

.79318 

.62274 

.78243 

.63630 

.77144 

29 

32 

.58118 

.81378' 

.59529 

.80351 

.60922 

.79300 

.62297 

.78225 

.63653 

.77125 

28 

33 

.58141 

.813611 

.59552 

.80334 

.60945 

.79282 

1 .62320 

.78206 

.63675 

.77107 

27 

34 

.58165 

.81344 

.59576 

.80316 

.60968 

.79264 

.62342 

.78188 

.63698 

.77088 

26 

35 

.58189 

.81327 

.59599 

.80299: 

.60991 

.79247 

.62365 

.78170 

.63720 

.77070 

25 

36 

.58212 

.81310 

.590221 

.80282; 

.61015 

.79229 

.62388 

.78152 

.63742 

.77051 

24 

37 

.58236 

.81293 

.59046 

.80264! 

.61038 

.79211 

.62411 

.78134 

.63765 

.77033 

23 

38 

.58200 

.81276 

.59009 

.80247 

.61001 

.79193 

.62433 

.78116 

.63787 

.77014 

22 

39 

.58283 

.81259 

.59093 

.80230 

.61084 

.79176 

.02450 

.78098 

.63810 

.76996 

21 

40 

.58307 

.81242 

.59710 

.80212 

.61107 

.79158 

.62479 

.78079 

.63832 

.76977 

20 

41 

.5a330 

.81225 

1 .59739 

.80195' 

.61130 

.79140' 

.62502 

.78001! 

.03854 

.76959 

19 

42 

.58354 

.81208 

.59763 

.80178 

.61153 

.79122: 

.62524 

.78043 

.63877 

.76940 

18 

43 

.58378, 

.811911 

.59780 

.80160 

.61176 

.79105 

.62547 

.78025 

.03899 

.76921 

17 

44 

.584011 

.811741 

.59809 

.80143 

.61199 

.79087 

.62570 

.78007 

.63922 

.76903 

16 

45 

.584251 

.811571 

.59832 

.80125 

.61222 

.79069 

.62592 

.77988 

.63944 

.76884 

15 

46 

.58449, 

.81140' 

.59856 

.80108 

.61245 

.79051 

.02615 

.77970 

.63966 

.76866 

14 

47 

.58472, 

.811231 

.59879 

.80091 1 

.61268 

.79033 

.62638 

.77952 

.63989 

.76847 

13 

48 

.58496 

.81106; 

.59902 

.80073' 

.61291 

.79016 

.62060 

.77934 

.04011 

.76828 

12 

49 

..58519 

.81089 

.59926 

.81)056 

.61314 

.78998 

.02083 

.77916 

.04033 

.76810 

11 

50 

.58543 

.81072 

.59949 

. 80038 j 

.61337 

.78980 

.62706 

.77897 

.64056 

.76791 

10 

61 

..5a507 

.81055 

.59972 

.80021' 

.61360 

.78962 

.62728 

.77879 

.64078 

.76772 

9 

62 

..58.590 

.81038 

.59995 

80003 

.61383 

.78944 

.62751 

.77801 

.64100 

.76754 

8 

53 

.58014 

.81021 

.00019 

.79986 

.61406 

.78926 

.62774 

.77843 

.64123 

.76735 

7 

54 

.58037 

.81004 

.60042 

.79908 

.61429 

.78908 

.62796 

.77824 

.64145 

.76717 

6 

55  , 

.58061 

.80987 

.60005 

.79951 

.61451 

.78891 

.62819 

.77800 

.64167 

.76698 

5 

66 

.58684 

.80970 

.60089 

.79934 

.61474 

.78873 

.62842 

.77788 

.04190 

.76679 

4 

57 

.58708 

.809.53 

.60112 

.79916 

.61497 

.788.55 

.62864 

.77709 

.64212 

.76661 

3 

58 

.58731 

.80936 

.601% 

.79899 

.61.520 

.78a37 

.62887 

.77751 

.64234' 

.76642 

2 

59 

..587.55 

.80919 

.601.58 

.79881 

.61.543 

.78819 

.62909 

.77733 

. 642.56 

.76623 

1 

60 

.58779 

.801X12 

.60182 

.79864 

.61.566 

.78801 

.62932 

.77715 

.64279 

.76604 

_0 

/ 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin  1 

Sine 

/ 

64° 

63° 

62° 

61° 

60° 

656 


SUR  VE  YING. 


TAliLE  W\.— Continued. 
Natural  Sines  and  Cosines. 


40° 

41° 

0 



1 43*  I 

44° 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

1 Sine 

( !osin  ' 

Sine 

Cosln 

0 

.61279 

.76604 

. 6,5606 

.T.Wl 

766913 

.74314 

. 6.8-21  K) 

7731:3.5 

.6916<5 

.710.31 

60 

1 

.61301 

' . 76586 

.6.5628 

.7.’>4.52 

.66935 

.74295 

.68221 

.7-3116 

.(i91Hr 

.71911 

.59 

2 

.61323 

. 76.567 

.6.56.50 

.7.5-1:13 

.669.56 

.741276 

.68212 

.7:3096 

.71891 

TA 

3 

.61316 

.70.548 

.6.5672 

.7.5414 

.66978 

.742.56 

.68261 

.73ir,V, 

.71873 

57 

4 

.64368 

.76,5.30 

.6.5691 

.7.5395 

.66999 

.742.37 

.68285 

.7.3056 

.69.519 

.71K\3 

56 

5 

.61390 

.76.511 

.6.5716 

.75.375 

.67021 

.74217 

.6h:«x> 

.7:10:56 

.69.570 

.718.33 

55 

G 

.61112 

.76192 

.657:18 

.7.5856 

.67013 

.74198 

.68.327 

.7.3016 

.69.591 

.71813 

.5-4 

7 

.61135 

.76473 

.6.5759 

.75:1.37 

.670(11 

.74178 

.6H;>19 

.72996 

.(39612 

.71792 

.5.3 

8 

.61157 

.761.55 

.6.5781 

.75318 

.67036 

.741.59 

.68.370 

.72970 

.(;'.)(;:3:i 

.71772 

52 

9 

.61179 

.764:16 

.6.5803 

.7.5299 

.67107 

.741.39 

.68391 

.729:57 

.(:;)(3.5i 

.717.52 

51 

10 

.61501 

.76417 

.65825 

.75280 

.67129 

1.74120 

.68412 

.72937 

.69675 

^ .71732 

50 

11 

.64.524 

.76398 

.6.51347 

.7.5261 

.671.51 

.74100 

.68434 

.72917 

.69606 

.71711 

40 

12 

.61.546 

.76.380 

.6.5369 

.7.5241 

.67172 

.74080 

.681.55 

.72897 

.69717 

.71691 

48 

13 

.61.568 

.76.361 

1 .65391 

.75222 

.67194 

.74061 

,68176 

.72877 

.69737 

.71671 

47 

11 

.64.590 

70312 

1 .6.5913 

.7.5203 

.67215 

.74011 

.68497 

.72857 

.69758 

.71650 

46 

15 

.61612 

.76323 

.659.3,5 

.75184 

.672.37 

.74022 

.68518 

.728.37 

.69779 

.716.30 

45 

16 

.616.35 

.76301 

.6.5956 

.75165 

.67258 

.74002 

.68539 

.72817 

.69800 

.71610 

44 

17 

.64657 

.76286 

.65978 

.75146 

.67280 

.7.398.3 

.68561 

.72797 

.69821 

.71590 

4.3 

18 

.61679 

.76267 

.66000 

.75128' 

i .67301 

.73963 

.6.8582 

.72777 

.69842 

.71.569 

42 

19 

.61701 

.76218 

.66022 

.75107 

i .67323 

.7.3911 

.68003 

.72757 

.69862 

.71.549 

41 

20 

.64723 

.76229 

.66041 

.75088 

.67344 

.73924 

.08024 

,72737 

.69883 

.71529 

40 

21 

.61746 

.76210 

.66066 

.75069 

I .67366 

.73904 

.0-8045 

.72717 

.69004 

.71508 

30 

22 

.61768 

.76192 

.66038 

.7.5050 

.67087 

.7.38a5 

.680(30 

.72097 

.69925 

.71488 

38 

23 

.61790 

.76173 

.63109 

.750.30 

.67409 

.73865 

.08688 

.72077 

.69946 

.71468 

87 

24 

.64812 

.76154 

.66131 

.75011 

.67430 

.73846 

.68709 

.72057 

.69900 

.71447 

36 

25 

.64834 

.761:35 

.681.33 

.74992 1 

.67452 

.73826 

.68730 

.72037 

.69987 

.71427 

a5 

26 

.61856 

.76116 

.66175 

.74973' 

.67473 

.73806 

.68751 

.72017 

.70008 

.71407 

S4 

27 

.64878 

.76097, 

.66197 

.74953 

.67493 

.73787 

.68772 

.72597 

.70029 

.71386 

a3 

28  1 

.64901 

.76078 

.68218 

.74934 

.67516 

.73767 

.68793 

.72.577 

.70049 

.71366 

'32 

29 

.64923 

. 76059 

.66210 

.74915 

.67.5.38 

.7.3747 

.68814 

.7'2.557 

.70070 

.71345 

31 

30 

.64945 

.76041 

.66262 

.74896 

.67559 

.73728 

.68835 

.72537, 

.70091 

.71325 

30 

31 

.64967 

.76022 

.66284 

.74876 

.67580 

.73708 

.68857 

.72517' 

.70112 

.71305 

29 

32 

.64989 

.76003 

. 63 108 

.74857 

.67692 

.73683 

.08878 

.72497 

70132 

.71284 

28 

33  I 

.65011 

.75934 

.66327 

.74833 

.67623 

.73669 

.63899 

.72477! 

.70153 

.71264 

27 

34  1 

.65033 

. 75965 

.66349 

.74818 

.67645 

.7.3649 

.63920 

.724571 

.70174 

.7124.3 

26 

35 

. 65055 

.75946, 

.66371 

.74799 

.67666 

.73329 

.68941 

.72437 

.70195 

.71223 

35 

36  I 

.65077 

.75927! 

.68393 

.74780 

.67688 

.73610 

.63962 

.7'2417! 

,70215 

.7im 

24 

37  1 

.65100 

.75903 

.66414 

.74760 

.67709 

.7.3590 

.68983 

.72397 

.70236 

.71182 

23 

38  1 

.65122 

.75889 

.66433 

.74741 

.67730 

73570 

.69004 

.72377 

.70257 

.71162 

22 

39  i 

.65144 

.75870 

.634.58 

.74722 

.67752 

.7-3551 

.69025 

.72357 

.70277 

.71141 

21 

40 

.65166 

.75851: 

.66480 

.74703^ 

,67773 

.73531 

.69046 

.72337 

.70298 

.71121 

20 

41 

.65188 

.75832 

.66501 

.74683 

.67795 

.73511 

.69067 

.72317 

.70319 

.71100 

19 

42 

.65210 

.75813' 

.66523 

.74634 

.67816 

.73491 

.69088 

.72297 

.703.39 

.71080  18 

43 

.65232 

.75794' 

.66545 

.74644 

.67837 

.73472 

.69109 

.72277 

.70360 

.71059  17 

44 

. 65254 

. 75775 

.68566 

.74625 

.67859 

.73452 

.69130 

.72257 

.70381 

.71039 

16 

45 

.65276 

.75756 

.66588 

.74606 

.67880 

.73432 

.69151 

.72236 

.70401 

.71019 

15 

46 

.65298! 

. 75738 i 

.66610 

.74586 

.67901 

.73413 

.69172 

.72216 

.70422 

.70998 

14 

47 

.65320 

.75719' 

.66632 

.74567 

.67923 

.73393 

.69193 

.72196 

.70443 

.70978 

13 

48 

.65342 

.75700 

.66653 

.74548 

.67944 

.73373 

.69214 

.72176 

.70463 

.70957 

12 

49 

.65.364 

.7.5630 

. 66675 

.74528 

.67965 

.73353 

.69235 

.72156 

.70484 

.709:17 

11 

50 

.65386 

.75661 

.66697 

. 74509 

.67987 

.73333 

.69256 

.72136 

.70505 

.70916 

10 

51 

.65408 

.75642 

.66718 

.74489! 

.68008 

.73314 

.69277 

.72116 

.70525 

.70896 

9 

52 

.65430 

.75623 

.66740 

.74470 

.68029 

.7.3294 

.69298 

.72095 

.70546 

.70875 

8. 

53 

.65452 

.75604 

.66762! 

.74451  i 

.680.51 

.7:3274 

.69319 

.72075 

.70567 

.70a55 

7 

54 

.65474 

' . 75585 

.66783: 

.744:31: 

.68072 

.7:3254 

.69340 

.720.55 

.70.587 

.70834 

6 

55 

.65496 

1.75.566! 

.66805! 

.74412 

.68093 

.73234 

.69:361 

.72035 

.70608! 

.70813 

5 

56 

.65518 

. 75.547  i 

.66827 

.74392 

.68115 

.7:3215 

.69382 

.72015 

.70628; 

.70793 

4 

57 

.65540 

. 75528 1 

.66848 

1.74373: 

.681:36 

.73195 

.69403 

.71995 

.70649! 

.70772 

3 

58 

.65562 

. 7.5.509  i 

.66870 

! .743.53 

.68157 

.73175 

.69424 

.71974 

.70670, 

.70752 

2 

59 

.6.5.584 

.75490 

.66891 

.743:34: 

.68179 

.73155 

.69445 

.719.54 

.70690! 

.70731 

1 

60 

.65606 

.7.5471 

.66913 

!. 74314 1 

.68200 

.731.35 

.69460 

.71934! 

.70711' 

.70711 

0 

/ 

Cosin 

Sine 

Cosin 

1 Sine  1 

Cosin 

Sine 

Cosin 

Sine  1 

Cosin  1 

Sine 

/ i 

49°  i 

1 48°  1 

47° 

46°  1 

45° 

TABLES. 


657 


TABLE  VII. 

Natural  Tangents  and  Cotangents. 


0“ 

] 

“ 1 

i ^ 

1“ 

! 3“ 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

i Tang 

Cotang 

$ 

0 

.00000 

Infinite. 

.01746 

57.2900 

.03492 

28.6363 

.05241 

19.0811 

.00029 

3437.75 

.01775 

56.3506 

.03521 

28.3994 

.05270 

18.9755 

59 

2 

.00058 

1718.87 

.01804 

55.4415 

.0.3550 

28.1664 

.05299 

18.8711 

58 

3 

.00087 

1145.92 

.01833 

54.5613 

.03579 

27.9372 

.05328 

18.7678 

57 

4 

.00116 

859.436 

.01862 

53.7086 

.03609 

27.7117 

.05357 

18.6656 

56 

5 

.00145 

687.549 

.01891 

52.8821 

.03638 

27.4899 

.05387 

18.5645 

55 

6 

.00175 

572.957 

.01920 

52.0807 

.03667 

27.2715 

.05416 

18.4645 

54 

7 

.00204 

491.106 

.01949 

61.3032 

.0.3696 

27.0566 

.05445 

18.3655 

53 

8 

.00233 

429.718 

.01978 

50.5485 

.03725 

26.8450 

.05474 

18.2677 

52 

9 

.00262 

381.971 

.02007 

49.8157 

.03754 

26.6367 

.05503 

18.1708 

61 

10 

.00291 

343.774 

.02036 

49.1039 

.03783 

26.4316 

.05533 

18.0750 

50 

11 

.00320 

312.521 

.02066 

48.4121 

.03812 

26.2296 

.05562 

17.9802 

49 

12 

.00349 

286.478 

.02095 

47.7395 

.03842 

26.0307 

1.05591 

17.8863 

48 

13 

.00378 

264.441 

.02124 

47.0853 

.03871 

25.8348 

.05620 

17.7934 

47 

14 

.00407 

245.552 

.02153 

46.4489 

.03900 

25.6418 

.05649 

17.7015 

46 

15 

.00436 

229.182 

.02182 

45.8294 

.03929 

25.4517 

.05678 

17.6106 

45 

16 

.00465 

214.858 

.02211 

45.2261 

.03958 

25.2644 

.05708 

17.5205 

44 

17 

.00495 

202.219 

.02240 

44.6386 

.03987 

25.0798 

.05737 

17.4314 

43 

18 

.00524 

190.984 

.02269 

44.0661 

.04016 

24.8978 

.05766 

17.3432 

42 

19 

.00553 

180.932 

.02298 

43.5081 

.04046 

24.7185 

.05795 

17.2558 

41 

20 

.00582 

171.885 

.02328 

42.9641 

.04075 

24.5418 

.05824 

17.1693 

40 

21 

.00611 

163.700 

.02357 

42.4335 

.04104 

24.3675 

.05854 

17.0837 

39 

22 

.00640 

1.56.259 

.02386 

41.9158 

.04133 

24.1957 

.05883 

16.9990 

38 

23 

.00669 

149.465 

.02415 

41.4106 

.04162 

24.0263 

.05912 

16.9150 

37 

24 

.00698 

143.237 

.02444 

40.9174 

.04191 

23.8593 

.05941 

16.8319 

36 

25 

.00727 

137.507 

.02473 

40.4358 

.04220 

23.6945 

.05970 

16.7496 

35 

20 

.00756 

132.219 

.02502 

39.9655 

.04250 

23.5321 

.05999 

16.6681 

34 

27 

.00785 

127.321 

.02531 

39.5059 

.04279 

23.3718 

.06029 

16.5874 

33 

28 

.00815 

122.774 

.02560 

39.0568 

.04308 

23.2137 

.06058 

16.5075 

32 

29 

.00844 

11 8.. 540 

.02589 

38.6177 

.04337 

23.0577 

.06087 

16.4283 

31 

SO 

.00873 

114.589 

.02619 

38.1885 

.04366 

22.9038 

.06116 

16.3499 

30 

31 

.00902 

110.892 

.02648 

37.7686 

.04395 

22.7519 

.06145 

16.2722 

29 

32 

.00931 

107.426 

.02677 

37.3579 

.04424 

22.6020 

.06175 

16.1952 

|28 

a3 

.00960 

104.171 

.02706 

36.9560 

.04454 

22.4541 

.06204 

16.1190 

!27 

34 

.00989 

101.107 

.02735 

36.5627 

.04483 

22.3081 

.06233 

16.0435 

26 

35 

.01018 

98.2179 

.02764 

36.1776 

.04512 

22.1640 

.06262 

15.9687 

25 

36 

.01047 

95.4895 

.02793 

35.8006 

.04541 

22.0217 

.06291 

15.8945 

24 

37 

.01076 

92.9085 

.02822 

35.4313 

.04570 

21.8813 

.06321 

15.8211 

23 

38 

.01105 

90.4633 

.02851 

35.0095 

.04599 

21.7426 

.06350 

15.7483 

22 

39 

.01135 

88.1436 

.02881 

34.71.51 

.04628 

21.60.56 

.06379 

15.6762 

21 

40 

.01164 

85.9398 

.02910 

34.3678 

.04658 

21.4704 

.06408 

15.6048 

20 

41 

.01193 

83.8435 

.02939 

34.0273 

.04687 

21.3369 

.06437 

15.5340 

19 

42 

.01222 

81.8470 

.02908 

33.6935 

.0-4716 

21.2049 

.06467 

15.4638 

18 

43 

.01251 

79.9434 

.02997 

33.3662 

.04745 

21.0747 

.06496 

15.3943 

17 

44 

.01280 

78.1263 

.03026 

as. 0452 

.04774 

20.9460 

.06525 

15.3254 

16 

45 

.01309 

76.3900 

.03055 

32.7303 

.04803 

20.8188 

.06554 

15.2571 

15 

46 

.Class 

74.7292 

.03084 

32.4213 

.04833 

20.6932 

.06584 

15.1893 

14 

47 

.01367 

73.1390 

.03114 

32.1181 

.04862 

20.5691 

.00613 

15.1222 

13 

48 

.01396 

71.6151 

.03143 

31.8205 

.04891 

20.4465 

.06642 

15.0557 

12 

49 

.01425 

70.1533 

.03172 

31.5284 

.04920 

20.32.53 

.06671 

14.9898 

11 

50 

.01455 

68.7501 

.03201 

31.2416 

.04949 

20.2056 

.00700 

14.9244 

10 

51 

.01484 

67.4019 

.03230 

30.9599 

.04978 

20.0872 

.06730 

14.8596 

9 

52 

.01513 

66.10.55 

.03259 

30.6833 

.05007 

19.9702 

.06759 

14.7954 

8 

53 

.01542 

61.8580 

.03288 

30.4116 

.05037 

19.a546 

.06788 

14.7317 

7 

54 

.01.571 

63.6.567 

.03317 

30.1446 

.05066 

19.7403 

.06817 

14.6685 

6 

55 

.OlfXK) 

62.4992 

.aa346 

29.8823 

.05095 

19.6273 

.06847 

14.60.59 

5 

56 

.01629 

61.. 3829 

.03376 

29.0245 

.05124 

19.51.56 

.06876 

14.54;i8 

4 

57 

.016.58 

60.30.58 

.03405 

29.. 3711 

.05153 

19.40.51 

.06905 

14.4823 

3 

58 

.01687 

.59. 26.59 

.03434 

29.1220 

.05182 

19.29.59 

.069.34 

14.4212 

2 

59 

.01716 

58.2612 

.03463 

28.8771 

.05212 

19.1879 

.06963 

14.3607 

1 

60 

.01746 

57.2900 

.03492 

28,6.363 

.05241 

19.0811 

.00993 

14.3007 

0 

Cotang 

Tang 

Cotang 

Tang 

Cctnng 

Tang 

Cotang  1 

Tang 

/ 

89“ 

88“  1 

87“ 

86“ 

658 


SUR  VE  YING. 


TABLE  VW.  — Coutinued. 

Natural  Tangents  and  Cotangents. 


4° 

5“ 

e 

7 

Tang 

Cotang 

Tang 

Cot  an  g 

Tang 

Cotang 

Tang  1 Cotang 

0 

.06993 

14.:10()7 

.08749 

11 ,4:«)1 

.10510  1 

9.514:36 

.12278 

8.144.-15 

1 

.07022 

14.2411 

.08778 

11  ..3919 

.10.540 

9.48781 

.12:308 

8.12181 

59 

2 

.07051 

14.1821  1 

.0Kh07 

ll.:3;>40 

.10.569 

9.46141 

.123-38 

8.10.5.36 

58 

8 

.07080 

]4.12;i5  1 

.088:17 

11.3163 

.10.599 

9.4:3515 

.12367 

8.08600 

.57 

4 

.07110 

14.0655  1 

.08866 

11.2789 

.10628 

9.40'K)1 

.12:397 

8.06674 

.56 

5 

.07139 

14.0079 

.08895 

11.2117 

1 .10557 

9.38307  , 

.12426 

8.047.56 

.\5 

0 

.07168 

13.9507 

.08925 

11.2048 

' .10687 

9.35724  ! 

.12456 

8.02818 

.54 

7 

.07197 

13.8940  ! 

.089.54 

11.16.81 

1 .10716 

9.. 33 1.55 

.12485 

8.00948 

.53 

8 

.07227 

13.8378 

.08983 

11.1316 

.10746 

9.30.599 

.12515 

7.990.58 

52 

9 

.07256 

13.7821 

.OIK)  13 

11.09.54 

.1077'5 

9.28058 

.12.514 

7.97176 

51 

10 

.07285 

13.7267 

,09042 

11.0594 

.10805 

9.25530 

.12574 

7.9.536.2 

50 

11 

.07314 

13.6719 

.09071 

11.0237 

.10R-V4 

9.23016 

.12608 

7. 9.34.38 

49 

12 

.07:144 

13.6174  , 

.09101 

10.9882 

.10853 

9.20516 

.126.^3 

7.91.5H2 

48 

13 

.07373 

13.56:i4  ! 

.091 :10 

10,9.529 

.10893 

9.18028 

.12662 

7.897.34 

47 

14 

.07402 

13.5098 

1 .091.59 

10.9178 

.10922 

9.15.5.54 

.12692 

7.87895 

46 

15 

.07431 

13.4566 

.09189 

10.8829 

.10952 

9.13093 

.12722 

7.86004 

45 

IG 

.07461 

13.4039 

.09218 

10. m3 

.10981 

9.106-16 

.12751 

7.84242 

44 

17 

.07490 

13.3515 

.09217 

10.8139 

.11011 

9.08211 

.12781 

7.82128 

43 

18 

.07519 

13.2996 

.09277 

10.7797 

.11040 

9.0.5789 

.12810 

7.80622 

42 

19 

.07548 

13.2480 

.09.306 

10.74.57 

.11070 

9.03.379 

.12840 

7.78825 

41 

20 

.07578 

13.1969 

.09335 

10.7119 

.11099 

9.00983  1 

.12869 

7.770:i5 

40 

21 

.07607 

13.1461 

.09.365 

10.6783 

.11128 

8.98598  ! 

.12899 

7.7.5254 

39 

22 

.07636 

13.0958 

.09394 

10.6450 

.111.58 

8.96227 

.12929 

7.7.-1480 

38 

23 

.07665 

13.0458 

.09423 

10.6118 

.11187 

8.9.3867  1 

.12958 

7.71715 

.37 

24 

.07695 

12.9902 

.09453 

10.5789 

.11217 

8.91520 

.12988 

7.69957 

.36 

25 

.07724 

12.9469 

.09482 

10.5462 

.11246 

8.89185 

.13017 

7.68208 

.35 

26 

.07753 

12.8981 

.09511 

10.51.36 

.11276 

8.86862  ! 

.13047 

7.66466 

34 

27 

.07782 

12.8496 

.09541 

10.4813 

.11305 

8.84551  1 

.13076 

7.647.32 

.33 

28 

.07812 

12.8014 

.09570 

10.4491 

.11335 

8.82252 

.13106 

7.&3005 

32 

29 

.07841 

12.7536 

.09600 

10.4172 

.11364 

8.79964  ' 

.13136 

7.61287 

31 

30 

.07870 

12.7062 

.09629 

10.3854 

.11394 

8.77689 

.|13165 

7.59575 

30 

31 

.07899 

12.6591 

.09658 

10.3538 

.11423 

8.75425  1 

.13195 

7.57872 

29 

82 

.07929 

12.6124 

.09688 

10.3224 

.11452 

8.73172  ' 

.1:3224 

7.56176 

28 

33 

.07958 

12.5660 

.09717 

10.2913 

.11482 

8.70931 

.1:3254 

7.54487 

27 

34 

.07987 

12.5199 

.09746 

10.2602 

.11511 

8.68701 

.1.3284 

7.52806 

26 

35 

.08017 

12.4742 

.09776 

10.2294 

.11541 

8.66482 

.1.3313 

7.511.32 

25 

36 

.08046 

12.4288 

.09805 

10.1988 

.11570 

8.64275 

.1.3343 

7.49465 

24 

37 

.08075 

12.3838 

.09834 

10.1683 

.11600 

8.62078 

.ia372 

7.47806 

23 

38 

.08104 

12.3390 

.09864 

10.1.381 

.11629 

8.59893 

.1.3402 

7.46154 

22 

39 

.08134 

12.2946 

.09893 

10.1080 

.11659 

8.57718 

.13432 

7.44509 

I2I 

40 

.08163 

12.2505 

.09923 

10.0780 

.11688 

8.55555 

.13461 

7.42871 

20 

41 

.08192 

12.2067 

.09952 

10.0483 

.11718 

1 53402 

.13491 

7.41240 

19 

42 

.08221 

12.1632 

.09981 

10.0187 

.11747 

8.51259 

.13521 

7.39616 

18 

43 

.08251 

12.1201 

.10011 

9.98931 

.11777 

8.49128 

.13550 

7.37999 

17 

44 

.08280 

12.0772 

.10040 

9.96007 

.11806 

8.47007 

.13580 

7.36.389 

16 

45 

.08309 

12.0346 

,10069 

9.93101 

.11836 

8.44896 

.13609 

7.34786 

15 

46 

.08339 

11.9923 

.10099 

9.90211 

.11865 

8.42795 

.13639 

7.33190 

14 

47 

.08368 

11.9504 

.10128 

9.87338 

.11895 

8.40705 

.13669 

7.31600 

13 

48 

.08397 

11.9087 

.10158 

9.84482 

.11924 

8.38625 

.13698 

7.30018 

12 

49 

.08427 

11.8673 

.10187 

9.81641 

.11954 

8.36555 

.13728 

7.28442 

11 

60 

.08456 

11.8262 

.10216 

9.78817 

.11983 

8.34496 

.13758 

7.26873 

10 

51 

.08485 

11.7853 

.10246 

8.76009 

.12013 

8.32446 

.13787 

7.25310 

9 

52 

.08514 

11.7448 

.10275 

9.7.3217 

.12042 

8.. 30406 

.13817 

7.23754 

8 

63 

.08544 

11.7045 

.10305 

9.70441 

.12072 

8.28376 

.13846 

7.22204 

7 

64 

.08573 

11.6645 

.10:334 

9.67680 

.12101 

8.26355 

.1.3876 

7.20661 

6 

55 

.08602 

11.6248 

.10363 

9.649:35 

.12131 

8.24:345 

.13906 

7.19125 

5 

56 

.086:12 

11.5853 

.10393 

9.62205 

.12160 

8.22344 

.13935 

7.17594 

4 

57 

.08661 

11.5461 

.10422 

9.59490 

.12190 

8.20a52 

.13965 

7.16071 

3 

58 

.08690 

11  5072 

.10452 

9.56791 

.12219 

8.18370 

.13995 

7.14553 

2 

69 

.08720 

11.4685 

.10481 

9.54106 

.12249 

8.16:398 

.14024 

7.1.3042 

1 

60 

.08749 

11.4:101 

.10510 

9. 51 436 

.12278 

8.1 44^5 

.14054 

7.115.37 

0 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

1 Cotang 

Tang 

85» 

0 

CO 

83° 

! 82° 

TABLES. 


659 


TABLE  VII. — Continued. 
Natural  Tangents  and  Cotangents. 


8° 

1 9“ 

>-* 

I 11“ 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

/ 

0 

.14054 

7.11537 

.15838 

6.31375 

.17633 

5.67128 

.19'438 

5.14455 

m 

1 

.14084 

7.10038 

.15868 

6.30189 

.17663 

5.66165 

.19468 

5.13658 

59 

2 

.14113 

7.08546 

.15898 

6.29007 

.17693 

5.65205 

.19498 

5.12862 

58 

3 

.14143 

7.07059 

.15928 

6.27829 

.17723 

5.64248 

.19.529 

5.12069 

57 

4 

.14173 

7.05579 

.15958 

6.26655 

.17753 

5.63295 

.195.59 

5.11279 

56 

5 

.14202 

7.04105 

.15988 

6.25486 

.17783 

5.62344 

.19589 

5.10490 

55 

6 

.14232 

7.02637 

.16017 

6.24321 

.17813 

5.61397 

.19019 

5.09704 

54 

7 

.14262 

6.91174 

.16047 

6.23160 

.17843 

5.60452 

.19649 

5.08921 

53 

8 

.14291 

6.99718 

.16077 

0.22003 

.17873 

5.59511 

.19680 

5.08139 

52 

9 

.14321 

6.98268 

.16107 

0.20851 

.17903 

5.58573 

.19710 

5.07360 

51 

10 

.14351 

6.96823 

.16137 

6.19703 

.17933 

5.57638 

.19740 

5.06584 

50 

11 

.14381 

6.95385 

.16167 

0.18.559  1 

.17963 

5.56706 

.19770 

5.05809 

49 

12 

.14410 

6.93952 

.16196 

0.17419 

.17993 

5.55777 

.19801 

5.05037 

48 

13 

.14440 

6.92525 

.16226 

6.16283 

.18023 

5.54851 

.19831 

5.042G7 

47 

14 

.14470 

6.91104 

.16256 

6.15151 

.18053 

5.53927 

.19861 

5.03499 

46 

15 

.14499 

6.89688 

.16286 

6.14023 

.18083 

5.53007 

.19891 

5.02734 

45 

16 

.14529 

6.88278 

.16316 

6.12899 

.18113 

5.52090 

.19921 

5.01971 

44 

17 

. 14559 

6.86874 

.10346 

6.11779 

.18143 

5.51176 

.19952 

5.01210 

43 

18 

.14588 

6.85475 

.16376 

6.10664 

.18173 

5.50264 

.19982 

5.00451 

42 

19 

.14618 

6.84082 

.16405 

6.09552 

.18203 

5.49356 

.20012 

4.99695 

41 

20 

.14648 

6.82694 

.16435 

6.08444 

.18233 

5.48451 

.20042 

4.98940 

40 

21 

.14678 

6.81312 

.16465 

6.07340 

.18263 

5.47548 

.20073 

4.9S188 

39 

22 

.14707 

6.79936 

.16495 

0.06240 

.18293 

5.40048 

.20103 

4.97438 

38 

23 

.14737 

6.78564 

.16525 

C. 05143 

.18323 

5 . 4.JI  51 

.20133 

4.96690 

37 

24 

.14767 

6.77199 

.16555 

6.04051 

.18353 

5.44857 

.20104 

4.95945 

36 

25 

.14796 

6.75838 

.16585 

6.02962 

.18384 

5.43966 

.20194 

4.95201 

35 

26 

.14826 

6.74483 

.16615 

6.01878 

.18414 

5.43077 

.20224 

4.94460 

.34 

27 

.14856 

6.73133 

.16645 

6.00797 

.18444 

5.42192 

.20254 

4.93721 

33 

28 

.14886 

6.71789 

.16674 

5.99720 

.18474 

5.41309 

.20285 

4.92984 

32 

29 

.14915 

6.70450 

.16704 

5.93046 

.18504 

5.40429 

.20315 

4.92249 

31 

30 

.14945 

6.69116 

.16734 

5.97576 

.18534 

5.39552 

.20345 

4.91516 

30 

31 

.14973 

6.67787 

.16764 

5.96510 

.18564 

5.38677 

.20376 

4.90785 

29 

32 

.15005 

6.66463 

.16794 

5.95448 

.18594 

5.37805 

.20406 

4.90056 

28 

33 

.15034 

6.65144 

.16824 

5.94390 

.18624 

5.36936 

.20436 

4.89330 

27 

34 

.15064 

6.63831 

.16854 

5.93335 

.18054 

5.36070 

.20466 

4.88605 

20 

35 

.15094 

6.62523 

.16884 

5.92283 

.18684 

5.35206 

.20497 

4.87882 

25 

36 

.15124 

6.61219 

.16914 

5.91236 

.18714 

5.34345 

.20527 

4.87162 

24 

37 

.15153 

6.59921 

.16944 

5.90191 

.18745 

5.33487 

.20557 

4.86444 

23 

38 

.15183 

6.58627 

.16974 

5.89151 

.18775 

5.32631 

.20588 

4.85727 

22 

39 

.15213 

6.57339 

.17004 

5.88114 

.18805 

5.31778 

.20618 

4.85013 

21 

40 

.15243 

6.56055 

.17033 

5.87080 

.18835 

5.30928 

.20648 

4.84300 

20 

41 

.15272 

6.54777 

.17063 

5.86051 

.18865 

5.. 30080 

.20679 

4.83590 

19 

42 

.15302 

6.53503 

.17093 

5.85024 

.18895 

5.29235 

.20709 

4.82882 

18 

43 

.15332 

6.52234 

.17123 

5.84001 

.18925 

5.28393 

.20739 

4.82175 

17 

44 

.15362 

6.50970 

.17153 

5.82982 

.18955 

5.275.53 

.20770 

4.81471 

16 

45 

.15391 

6.49710 

.17183 

5.81966 

.18986 

5.26715 

.20800 

4.80769 

15 

46 

.15421 

6.48456 

.17213 

5.80953 

.19016 

5.25880 

.20830 

4.80068 

14 

47 

.15451 

6.47206 

.17243 

5.79944 

.19046 

5.25048 

.20861 

4.79370 

13 

48 

.15481 

6.45961 

.17273 

5.78938 

.19076 

5.24218 

.20891 

4.78673 

12 

49 

.15.511 

6.44720 

.17303 

5.77936 

.19106 

5.2.3391 

.20921 

4.77978 

11 

50 

.15640 

6.43484 

.17333 

5.76937 

.19136 

5.22566 

.20952 

4.77286 

10 

51 

.1.5.570 

6.422.53 

.17303 

5.75941 

.19166 

5.21744 

.20982 

4.76595 

9 

52 

.15600 

6.41020 

.17393 

5.74949 

.19197 

5.20925 

.21013 

4.75906 

8 

53 

.156.30 

6.39804 

.17423 

5.73960 

.19227 

5.20107 

.21043 

4.75219 

7 

54 

.15660 

6.3a587 

.17453 

5.72974 

.19257 

5.19293 

.21073 

4.74534 

6 

55 

.15689 

6.. 37.374 

.17483 

5.71992 

.19287 

5.18480 

.21104 

4.73851 

5 

56 

.1.5719 

6.. 361 65 

.17513 

5.71013 

.19317 

5.17071 

.21134 

4.73170 

4 

57 

.15749 

6.34961 

.17543 

5.70037 

.19347 

5.16863 

.21164 

4.72490 

3 

58 

.15779 

6.. 3.3761 

.17573 

5.69064 

.19378 

5.160.58 

.21195 

4.71813 

2 

59 

.15809 

6.. 32.566 

.17603 

5.68094 

.19408 

5.1.52.50 

.21225 

4.71137 

1 

60 

.1.58.38 

6.31375 

.17033 

5.67128 

.19438 

5.144.55 

.21256 

4.70463 

0 

/ 

Co  tang 

Tang 

Cotang 

Tang 

Cotang 

Tang  1 

Cotang 

Tang 

/ 

81“ 

1 80“  1 

79“  ' 

78“ 

66o 


SUK  VE  YING. 


TABLE  VII,  — Con ti)i Kcd. 
Natural  Tangents  and  Cotan(;ents. 


12° 

13°  1 

14° 

15° 

Tanpr 

Cotang 

Tang 

Cotang  1 

Tang 

' Cotang 

Tang 

1 Cotang 

r 

0 

.2125(5 

4.7046:1 

.23087 

4.:i3r48  , 

.249:1.3 

r-i.uiW 

.26795 

60 

1 

.21286 

4.69791 

.23117 

4.32573 

.24964 

4.00.5H2 

.26826 

3.72771 

59 

2 

.21316 

4.69121 

.23148 

4.32001 

.24995 

4.000H(; 

.2(5857 

; 3.72338 

68 

3 

.21347 

4.684.52 

.2.3179 

4.311:10 

.25026 

3.99.592 

.26888 

3.71907 

67 

4 

.21377 

4.67786 

.2:}209 

4.30860 

.250.56 

3.99099 

.26920 

, 3.71476 

66 

6 

.21408 

4.67121 

.2:1240 

4.30291 

.25087 

3.9(3(507 

.2(/951 

3.71016 

55 

6 

.21438 

4.66458 

.2:1271 

4.29724 

.25118 

3.98117 

.26982 

3.70616 

64 

7 

.21469 

4.65797 

.2:1.301 

4.29159 

.25149 

3.97627 

.27013 

3. 701 88 

63 

8 

.21499 

4.651.38 

.23332 

4.28595 

.25180 

3.97139 

.27(M4 

3.69761 

62 

« 

.215^9 

4.61480 

.23363 

4.280;i2 

.25211 

3.96651 

.27(^6 

3.69.335 

51 

10 

.21560 

4.63825 

.23393 

4.27471 

.25212 

3.96165 

.27107 

3.08909 

50 

11 

.21590 

4.6.3171 

.23124 

4.26911 

^ .2.5273 

3.9.56.S0 

.27138 

3.CR4R.5 

49 

12 

.21621 

4.62518 

.23455 

4.263.52 

1 .25.304 

3.95196 

.271(59 

3.(5i;0(51 

48 

13 

.21651 

4.61868 

.2:1485 

4.2.5795 

1 .2.5.235 

3.91713 

.27201 

3.(57(;  .8 

47 

14 

.21682 

4.61219 

.2.3516 

4.25239 

1 .25.366 

3.942.32 

.272.52 

t 3.(57217 

i46 

15 

.21712 

4.60.572 

.21517 

4.24685 

' .25.397 

3.9.3751 

.27263 

3.(56796 

45 

16 

.21743 

4.59927 

.21578 

4.241.32 

.25428 

3.9.3271 

.27294 

3.66376 

44 

17 

.21773 

4.. 59283 

.2.3608 

4.23580 

.254.59 

3.1r2793 

!2.'32G 

3.6.5957 

43 

18 

.21804 

4.58641 

.2:1039 

4.23630 

.25190 

3.92316 

27.357 

3.65538 

42 

ly 

.21834 

4.58001 

.23070 

4.22481 

.25521 

3.91839 

! 27388 

3.6.5121 

41 

20 

.21864 

4.57363 

.23700 

4.21933 

.25552 

3.91304 

.27419 

3.04705 

40 

21 

.21895 

4.56726 

.23731 

4.21387 

.25583 

3.90890 

.27451 

3.64289 

39 

22 

.21925 

4.56091 

.23762 

4.20f«2 

.2.5614 

3.90417 

.27482 

3.03874 

38 

23 

.21956 

4.55458 

.23793 

4.20298 

.2r>645 

3.89945 

.27513 

3.6.3461 

37 

24 

.21986 

4.64826 

.23823 

4.19756 

.25676 

3.89474 

.27515 

3.63048 

36 

25 

.22017 

4.54196 

.23354 

4.19215 

.25707 

3.89004 

.27576 

3.62636 

35 

26 

.22047 

4.53568 

.23885 

4.18675 

.257.38 

3.88536 

.27607 

3.622^ 

34 

27 

.22078 

4.02941 

.23916 

4.181.37 

.2.5769 

3.880C8 

.276:18 

[ 3.61814 

a3 

28 

.22108 

4.52316 

.23946 

4.17600 

2.5800 

3.87601 

.27670 

1 3.61405 

32 

29 

.22139 

4.51093 

.23977 

4.17064 

; 25831 

3.87136 

.27701 

1 3.60996 

31 

30 

.22169 

4.51071 

.24008 

4.16530 

.25862 

3.86671 

.27732 

1 3 C0588 

30 

31 

.22200 

4.50451 

.24039 

4.15997 

.25893 

3.86208 

.27764 

3.C0181 

29 

32 

.22231 

4.49832 

.24069 

4.15465 

.2.5924 

3.85745 

.27795 

3.59770 

23 

33 

.22261 

4.49215 

.24100 

4.14934 

.25955 

3.85284 

.27826 

3.59370 

27 

34 

.22292 

4.48600 

.24131 

4.14405 

.2.5986 

3.84824 

.278.58 

3.589G6 

26 

35 

.22322 

4.47986 

.24162 

4.13877 

.26017 

3.84364 

.27889 

3.58562 

25 

36 

.22353 

4.47374 

.24193 

4.13350 

.26048 

3.83906 

.27921 

3.. 581 60 

24 

37 

.22383 

4.46734 

.24223 

4.12825 

.26079 

3.83449 

.27952 

3.57758 

23 

38 

.22414 

4.46155 

.24254 

4.12301 

.26110 

3.82992 

.27983 

3.57357 

22 

39 

.22444 

4.45548 

.24285 

4.11778 

.26141 

3.82537 

.28015 

3.56957 

21 

40 

.22475 

4.44942 

.24316 

4.11256 

.26172 

3.82083 

.28046 

3.56557 

20 

41 

.22505 

4.44338 

.24347 

4.10736 

.26203 

3.81630 

.28077 

3.56159 

19 

42 

.22536 

4.43735 

.24377 

4.10216 

.262.35 

3.81177 

.28109 

3.55761 

18 

43 

.22567 

4.43134 

.24408 

4.09699 

.26286 

3.80726 

.28140 

3.55364 

17 

44 

.22597 

4.42534 

.24439 

4.09182 

.26297 

3.80276 

.28172 

3.54968 

IG 

45 

.22628 

4.41936 

.24470 

4.08666 

.26328 

3.79827 

.28203 

3.54573 

15 

46 

.22658 

4.41340 

.24501 

4.08152 

.26359 

3.79378 

.28234 

3..5417'9 

14 

47 

.22689 

4.40745 

.24532 

4.07639 

.28390 

3.78931 

.23266 

3.53765 

13 

48 

.22719 

4.40152 

.24562 

4.07127 

.26421 

3.78485 

.28297 

3.53393 

12 

49 

.22750 

4.39560 

.24593 

4.06616 

.26452 

3.78040 

.28329 

3.53001 

11 

50 

.22781 

4.38969 

.24624 

4.06107 

.26483 

3.77595 

.28360 

3.52609 

10 

51 

.22811 

4.38381 

.24655 

4.05599 

.26515 

3.77152 

.28391 

3.52219 

9 

52 

.22842 

4.37793 

.24686 

4.05092 

.26546 

3.76709 

.28423 

3.51829 

8 

53 

.22872 

4.37207 

.24717 

4,04586 

.26577 

3.76268 

.28454 

3.51441 

7 

54 

.22903 

4.36623 

.24747 

4.04081 

.26608 

3.75828 

.28486 

3.51053 

6 

55 

.22934 

4.36040 

.24778 

4.03578 

.26639 

S. 75388 

.28517 

3.50G66 

5 

56 

.22964 

4.35459 

.24809 

4.03076 

.26670 

3.74950 

.28549 

3.. 50279 

4 

57 

.22995 

4.31879 

.24840 

4.02574 

.26701 

3.74512 

.28580 

3.49894 

3 

58 

.23026 

4.34300 

.24871 

4.02074 

.26733 

3.74075 

.28612 

3.49509 

2 

59 

.23056 

4.3:1723 

.24902 

4.01576 

3.73640 

.28643 

3.49125 

1 

ra 

.23087 

4.33148 

.24933 

4.01078 

.26795 

3.73205 

_^28675 

3.48741 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

77° 

1 76° 

75° 

740 

TABLES. 


66l 


TABLE  VTI. — Continued. 
Natural  Tangents  and  Cotangents. 


16“ 

17° 

oo 

0 

19° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

9 

0 

.28675 

3.48741 

.30573 

3.27085 

■ .32492 

3.07768 

.34433 

2.90421 

1 

.28706 

3.48359 

.30605 

3.26745 

.32524 

3.07464 

.34465 

2.90147 

59 

2 

.28738 

3.47977 

.30637 

3.26406 

.32556 

3.07160 

.34498 

2.89873 

58 

3 

.28769 

3.47596 

.30669 

3.26067 

.32588 

3.06857 

.34530 

2.89600 

57 

4 

.28800 

3.47216 

.30700 

3.25729 

.32621 

3.06554 

.34563 

2.89327 

56 

5 

.28832 

3.46837 

.30732 

3.25392 

.32653 

3.06252 

.34596 

2.89055 

55 

6 

.28864 

3.46458 

.30764 

3.25055 

.32685 

3.05950 

.34628 

2.88783 

54 

7 

.28895 

3.46080 

.30796 

3.24719 

.32717 

3.05649 

.34661 

2.88511 

53 

8 

.28927 

3.45703 

.30828 

3.24383 

.32749 

3.05349 

.34693 

2.88240 

52 

9 

.28958 

3.45327 

.30860 

3.24049 

.32782 

3.05049 

.34726 

2.87970 

51 

10 

.28990 

3.44951 

.30891 

3.23714 

.32814 

3.04749 

.34758 

2 87700 

50 

11 

.29021 

3.44576 

.30923 

3.23381 

.32846 

3.04450 

.34791 

2.87430 

49 

12 

.29053 

3.44202 

.30955 

3.23048 

.32878 

3.04152 

.34824 

2.87161 

48 

13 

.29084 

3.43829 

.30987 

3.22715 

.32911 

3.03854 

.34856 

2.86892 

47 

14 

.29116 

3.43456 

.31019 

3.22384 

.32943 

3.03556 

.34889 

2.86624 

46 

15 

.29147 

3.43084 

.31051 

3.22053 

.32975 

3.03260 

.34922 

2.86356 

45 

16 

.29179 

3.42713 

.31083 

3.21722 

.33007 

3.02963 

.34954 

2.86089 

44 

17 

.29210 

3.42343 

.31115 

3.21392 

.33040 

3.02667 

.34987 

2.85822 

43 

IS 

.29242 

3.41973 

.31147 

3.21063 

.33072 

3.02372 

.35020 

2.85555 

42 

19 

.29274 

3.41604 

.31178 

3.20734 

.33104 

3.02077 

.35052 

2.85289 

41 

20 

.29305 

3.41236 

.31210 

3.20406 

.33136 

3.01783 

.35085 

2.85023 

40 

21 

.29337 

3.40869 

.31242 

3.20079 

.33169 

3.01489 

.35118 

2.84758 

39 

22 

.29368 

3.40502 

.31274 

3.19752 

.33201 

3.01196 

.35150 

2.84494 

38 

23 

.29400 

3.40136 

.31306 

3.19426 

.33233 

8.00903 

.35183 

2.84229 

37 

24 

.29432 

3.39771 

.31338 

3.19100 

.33266 

3.00611 

.35216 

2.83965 

36 

25 

.29463 

3.39406 

.31370 

3.18775 

.33298 

3.00319 

.35248 

2.83702 

35 

26 

.29495 

3.39042 

.31402 

3.18451 

.33330 

3.00023 

.35281 

2.83439 

34 

27 

.29526 

3.38679 

.31434 

3.18127 

.33363 

2.99738 

.35314 

2.83176 

33 

28 

.29558 

3.38317 

.31466 

3.17804 

.33395 

2.99447 

.35346 

2.82914 

32 

29 

.29590 

3.37955 

.31498 

3.17481 

.33427 

2.99158 

.35379 

2.82653 

31 

30 

.29621 

3.37594 

.31530 

3.17159 

.33460 

2.98868 

.35412 

2.82391 

30 

31 

.29653 

3.37234 

.31562 

3.16838 

.33492 

2.98580 

.35445 

2.82130 

29 

32 

.29685 

3.36875 

.31594 

3.16517 

.33524 

2.98292 

.35477 

2.81870 

28 

33 

.29716 

3.36516 

.31626 

3.16197 

.33557 

2.98004 

.35510 

2.81610 

27 

34 

.29748 

3.36158 

.31658 

3.15877 

.33589 

2.97717 

.35543 

2.81350 

26 

35 

.29780 

3.35800 

.31690 

3.15558 

.33621 

2.97430 

.35576 

2.81091 

25 

36 

.29811 

3.35443 

.31722 

3.15240 

.33654 

2.97144 

.35608 

2.80833 

24 

37 

.20843 

3.35087 

.31754 

3.14922 

.33686 

2.96858 

.35641 

2.80574 

23 

38 

.29875 

3.34732 

.31786 

3.14605 

.33718 

2.96573 

.35674 

2.80316 

22 

39 

.29906 

3.34377 

.31818 

3.14288 

.33751 

2.96288 

.35707 

2.80059 

21 

40 

.29938 

3.34023 

.31850 

3.13972 

.33783 

2.96004 

.35740 

2.79802 

20 

41 

.29970 

, 3.33670 

.31882 

3.13656 

.33816 

2.95721 

.35772 

2.79545 

19 

42 

.30001 

3.33317 

.31914 

3.13341 

.33848 

2.95437 

.35805 

2.79289 

18 

43 

.30033 

3.329G5 

.31946 

3.13027 

.33881 

2.95155 

.35838 

2.79033 

17 

44 

.30065 

3.32614 

.31978 

3.12713 

.33913 

2.94872 

.35871 

2.78778 

16 

45 

.30097 

3.32264 

.32010 

3.12400 

.a3945 

2.94591 

.35904 

2.78523 

15 

46 

.30128 

3.31914 

.32042 

3.12087 

.33978 

2.94309 

.35937 

2.78269 

14 

47 

.30160 

3.31565 

.32074 

3.11775 

.34010 

2.94028 

.35969 

2.78014 

13 

4C 

.30192 

3.31216 

.32106 

3.11464 

.34043 

2.93748  ' 

.36002 

2.77761 

12 

49 

.30224 

3.30868 

.32139 

3.11153 

.34075 

2.93468 

.36035 

2.77507 

11 

50 

.30255 

3.30521 

.32171 

3.10842 

.34108 

2.93189 

.36068 

2.77254 

10 

51 

.30287 

3.30174 

.32203 

3.10532 

.34140 

2.92910 

.36101 

2.77002 

9 

.52 

.30319 

3.29829 

.32235 

3.10223 

.34173 

2.92632 

.36134 

2.76750 

8 

53 

.30:i51 

3.29483 

.32267 

3.09914 

.34205 

2.92354 

.36167 

2.76498 

7 

54 

.30382 

3.29139 

.32299 

3.09606 

.34238 

2.92076 

.36199 

2.76247 

6 

55 

.30414 

3.28795 

.32331 

3.09298 

.34270 

2.91799 

.36232 

2.75996 

5 

56 

.30446 

3.28452 

.32363 

3.08991 

.34303 

2.91523 

.36265 

2.75746 

4 

57 

.30478 

3.28109 

.32396 

3.08685 

.34335 

2.91246 

.36298 

2.75496 

3 

58 

.30509 

3.27767 

.32428 

3.0a379 

.34368 

2.90971 

.36331 

2.75246 

2 

59 

.30541 

3.27426 

.32460 

3.08073 

.34400 

2.90696 

.36364 

2.74997 

1 

60 

.30573 

3.27085 

_^32492 

3.07768 

.34433 

2.90421 

.36397 

2.71718 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

f 

CO 

o 

to 

71° 

i 

o 

662 


SURVEYING. 


TABLE  VII.  — Con  tin  ned. 
Natural  Tangents  and  Cotangfcnts. 


1 20» 

21» 

22“ 

CO 

o 

1 Tanp: 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

1 Tang 

Cotang 

0 

.36397 

2.74748 

.38;i86 

.40403 

2.47.509 

1 .V24i7 

2.3558T 

60 

1 

.36430 

2.74499 

.31W20 

2.60283 

.404.36 

2.47302 

.42182 

2. 3.6395 

59 

2 

.36163 

2.742.51 

.384.53 

2.600.57 

.40470 

2.47095 

1 .42516 

2.85205 

.58 

3 

.36196 

2.74004 

.38487 

2.. 59831 

.40504 

2.46888 

.42,651 

2.. 650 15 

57 

4 

.36529 

2.737.56 

.a8520 

2.. 59606 

.40.5.'i8 

2.46682 

1 .42.585 

2. 61825 

56 

6 

.36562 

2.7.3509 

.38.5.53 

2.. 59381 

.40572 

2.46116 

.42619 

2,. 616.36 

.55 

6 

.36595 

2.73263 

.38.587 

2.591.56 

.40606 

2.46270 

.426.54 

2. 344 17 

54 

7 

.36628 

2.73017 

.38620 

2.. 5.89.32 

.40640 

2.46065 

.42688 

2.. 612.58 

53 

8 

.36661 

2.72771 

..386.54 

2.. 587  08 

.40674 

2.4.5.860 

.42722 

2. 64069 

52 

9 

! .36691 

2.72.526 

..186.87 

2.. 584  84 

.40707 

2.4.5655 

! 42757 

2..6‘1881 

51 

10 

; .36727 

2.72281 

.38721 

2.58261 

.40741 

2.45151 

.42791 

2.33693 

50 

11 

.36760 

2.720.36 

..38754 

2.580.38 

.40775 

2.4.5246 

.42826 

2.66505 

49 

12 

.36793 

2.71792 

.38787 

2.57815 

.40809 

2.45013 

.42860 

2.86317 

48 

13 

.36826 

2.71548 

..33821 

2.. 57.593 

.40843 

2.41839 

.42891 

2. 631 .30 

47 

14 

.368.59 

2.71305 

..18854 

2.57371 

.4087^ 

2.41636 

.42929 

2.. 32943 

46 

15 

.36892 

2.71062 

.3.8888 

2.. 571.50 

.40911 

2.414.33 

.42963 

2. 327.56 

45 

16 

.36925 

2.70819 

..38921 

2.. 56928 

.40915 

2.412.30 

.42998 

2. 32570 

44 

T7 

.36953 

2.70.577 

..189.55 

2.56707 

.40979 

2.41027 

.4.30.32 

2. 32.383 

43 

18 

.36991 

2.70335 

.38988 

2.. 56487 

.41013 

2.4.3825 

.4.3067 

2.. 321 97 

42 

19 

.37024 

2.70091 

.39022 

2.. 56266 

.41017 

2.4.3623 

.43101 

2. .32012 

41 

20 

.37057 

2.69853 

.39055 

2.56046 

.41081 

2.43*122 

.43136 

2.31826 

40 

21 

.37090 

2.69612 

.39089 

2.. 55827 

.41115 

2.4.3220 

.4.3170 

2.316*11 

39 

22 

.37123 

2.69371 

.39122 

2.53608 

.41149 

2.4.3019 

.43205 

2.31456 

38 

23 

.37157 

2.69131 

.39156 

2.55389 

.41183 

2. 42.81 9 

.4.3239 

2.31271 

37 

24 

.37190 

2.68892 

.39190 

2.55170 

.41217 

2.42618 

.43274 

2.31086 

36 

25 

.37223 

2.68653 

.39223 

2.. 54952 

.412.51 

2.42418 

.4.3.308 

2.. 30902 

35 

26 

.37256 

2.68414 

..39257 

2.54734 

.41285 

2.42218 

.4.3.343 

2.. 3071 8 

.34 

27 

.37289 

2.68175 

.39290 

2.54516 

.41319 

2.42019 

.43378 

2.. 30534 

83 

28 

.37322 

2 679,37 

.39324 

2.54299 

.41:3.53 

2.41819 

.4.3412 

2.. 38351 

32 

29 

.37355 

2.67700 

..39357 

2.. 540.82 

.41387 

2.41620 

.4.3447 

2.. 301 67 

31 

30 

.37383 

2.67462 

.39391 

2.53865 

.41421 

2.41421 

.43481 

2.29984 

30 

31 

.37422 

2.67225 

.39425 

2.53648 

.41455 

2.41223 

.43516 

2.29801 

29 

32 

.37455 

2.669S9 

.39458 

2.53432 

.41490 

2.41025 

.43550 

2.29619 

28 

33 

.37488 

2.66752 

.39492 

2.53217 

.41524 

2.40827 

.43585 

2.29437 

27 

34 

.37521 

2.66516 

.39526 

2.53001 

.41558 

2.40629 

.43620 

2.29254 

126 

35 

.37554 

2.66281 

.39559 

2.52786 

.41592 

2.404.32 

.43654 

2.29073  j 

25 

36 

.37588 

2.66046 

' .39593 

2.52571 

.41626 

2.402.35 

.43689 

2.28891  ^ 

24 

37 

.37621 

2.65811 

, .39626 

2.52357 

.41660 

2.40038 

.43724 

2.28710  1 

23 

38 

.37654 

2.65576 

1 .39660 

2.52142 

.41694 

2.39341 

.43758 

2.28528  1 

22 

39 

.37687 

2.65342 

' .39694 

2.51979 

.41728 

2.39645 

.43793 

2.28348  1 

21 

40 

.37720 

2.65109 

.39727 

2.51715 

.41763 

2.39449 

.43828 

2.28167  120 

41 

.37754 

2.64875 

.39761 

2.51502 

.41797 

2.39253 

.43862 

2.27987 

19 

42 

.37787 

2.64642 

.39795 

2.51289 

.41831 

2.39058 

.43897 

2.27806 

18 

43 

.37820 

2.64410 

..39329 

2.51076 

.41865 

2.38863 

.43932 

2.27626 

17 

44 

.37853 

2.64177 

.39862 

2.50364 

41899 

2.38663 

.43966 

2.27447 

16 

45 

.37887 

2.63945 

.39896 

2.50652 

.419^3 

2.38473 

.44001 

2.27267 

15 

46 

.37920 

2.63714 

.39930 

2.50440 

.41968 

2.38279 

.44036 

2.27088 

14 

47 

.379.53 

2.63483  i 

.39963 

2.50229 

.42002 

2.38084 

.44071 

2.26909 

13 

48 

.37986 

2.63252  I 

.39907 

2.50018 

.42036 

2.37891 

.44105 

2.26730 

12 

49 

.38020 

2.63021 

.40031 

2.49807 

.42070 

2.-37697 

.44140 

2.26552 

11 

50 

.38053 

2.62791 

.40065 

2.49597 

.42105 

2.37504 

.44175 

2.26374 

10 

51 

.38086 

2.62.561 

.40098 

2.49386 

.42139 

2.37311 

.44210 

2.26196 

9 

52 

.38120 

2.623;32 

.40132 

2.49177 

.42173 

2.37118 

.44214 

2.26018 

8 

53 

.38153 

2.62103 

.40166 

2.48967 

.42207 

2.36925 

.44279 

2.25840 

7 

54 

..38186 

2.61874 

.40200 

2.48758 

.42242 

2.. 36733 

.44314 

2.25663 

6 

55 

.38220 

2.61646 

.40234 

2.48549 

.42276 

2.36541 

.44349 

2.25486 

5 

56 

.38253 

2.61418 

.40267 

2 48340 

.42310 

2.36349 

.44384 

2.25309 

d 

57 

..38286 

2.61190 

.40.301 

2.48132 

.42345 

2.36158 

.44418 

2.25132 

3 

58 

.38.320 

2.60963 

.40335 

2.47924 

.42379 

2.35967 

.44453 

2.24956 

2 

59 

.38353 

2.60736 

.40369 

2.47716 

.42413 

2.35776 

.44488 

2 24780 

1 

60 

.38386 

2.60.509 

.40403 

2.47509 

.42447 

2.3.5.585 

.44523 

2.24604 

0 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang  1 

Tang 

69“ 

C5 

CO 

o 

67"  1 

60“ 

TABLES. 


663 


TABLE  VII.  — Continued. 

Natural  Tangents  and  Cotangents. 


24» 

25“ 

1 26“ 

27“ 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

/ 

0 

.44523 

2.24604 

.46631 

2.14451 

.48773 

2.05030 

.50953 

1.96261 

60 

1 

.44558 

2.24428 

.46666 

2.14288 

.48809 

2.04879 

.50989 

1.96120 

59 

2 

.44593 

2.24252 

.46702 

2.14125 

.48845 

2.04728 

.51026 

1.95979 

58 

3 

.44627 

2.24077 

.46737 

2.13963 

.48881 

2.04577 

.51063 

1.95838 

57 

4 

.44662 

2.23902 

.46772 

2.13801 

.48917 

2.04426 

.51099 

1.95698 

56 

5 

.44697 

2.23727 

.46808 

2.13639 

.48953 

2.04276 

.51136 

1.95557 

55 

6 

.44732 

2.23553 

.46843 

2.13477 

.48989 

2.04125 

.51173 

1.95417 

54 

7 

.44767 

2.23378 

.46879 

2.13316 

.49026 

2.03975 

.51209 

1.95277 

53 

8 

.44802 

2.23204 

.46914 

2.13154 

.49062 

2.03825 

.51246 

1.95137 

52 

9 

.44837 

2.23030 

.46950 

2.12993 

.49098 

2-.  0.3675 

.51283 

1.94997 

51 

10 

.44872 

2.22857 

.46985 

2.12832 

.49134 

2.03526 

.51319 

1.94858 

50 

11 

.44907 

2.22683 

.47021 

2.12671 

.49170 

2.0a376 

.51356 

1.94718 

49 

12 

.44942 

2.22510 

, .47056 

2.12511 

.49206 

2.03227 

.51393 

1.94579 

48 

13 

.44977 

2.22337 

! .47092 

2.12350 

.49242 

2.03078 

.51430 

1.94440 

47 

14 

.45012 

2.22164 

j .47128 

2.12190 

.49278 

2.02929 

.51467 

1.94301 

46 

15 

.45047 

2.21992 

1 .47163 

2.12030 

.49315 

2.02780 

.51503 

1.94162 

45 

16 

.45082 

2.21819 

.47199 

2.11871 

.49351 

2.02631 

.51540 

1.94023 

44 

17 

.45117 

2.21647 

.47234 

2.11711 

.49387 

2.02483 

.51577 

1.93885 

43 

18 

.45152 

2.21475 

.47270 

2.11552 

.49423 

2.02335 

.51614 

1.9,3746 

42 

19 

.45187 

2.21304 

.47305 

2.11392 

.49459 

2.02187 

.51651 

1.93608 

41 

20 

.45222 

2.21132 

.47341 

2.11233 

.49495 

2.02039 

.51688 

1.93470 

40 

21 

.45257 

2.20961 

.47377 

2.11075 

.49532 

2.01891 

.51724 

1.93332 

39 

22 

.45292 

2.20790 

.47412 

2.10916 

.49568 

2.01743 

.51761 

1.93195 

38 

23 

.45327 

2.20619 

.47448 

2.10758 

.49604 

2.01596 

.51798 

1.93057 

37 

24 

.45362 

2.20449 

.47483 

2.10600 

.49640 

2.01449 

.51835 

1.92920 

36 

25 

.45397 

2.20278 

.47519 

2.10442 

.49677 

2.01302 

.51872 

1.92782 

35 

26 

.45432 

2.20108 

.47555 

2.10284 

.49713 

2.01155 

.51909 

1.92645 

34 

27 

.45467 

2.19938 

.47590 

2.10126 

.49749 

2.01008 

.51946 

1.92508 

33 

28 

.45502 

2.19769 

.47626 

2.09969 

.49786 

2.00862 

.51983 

1.92371 

32 

29 

.45538 

2.19599 

.47662 

2.09811 

.49822 

2.00715 

.52020 

1.92235 

31 

30 

.45573 

2.19430 

.47698 

2.09654 

.49858 

2.00569 

.52057 

1.92098 

30 

31 

.45608 

2.19261 

.47733 

2.09498 

.49894 

2.00423 

.52094 

1.91962 

29 

32 

.45643 

2.19092 

.47769 

2.09341 

.49931 

2.00277 

.52131 

1.91826 

28 

33 

.45678 

2.18923 

.47805 

2.09184 

.49967 

2.00131 

.52168 

1.91690 

27 

34 

.45713 

2.18755 

.47840 

2.09028 

.50004 

1.99986 

.52205 

1.91554 

26 

35 

.45748 

2.18587 

.47876 

2.08872 

.50040 

1.99841 

.52242 

1.91418 

25 

36 

.45784 

2.18419 

.47912 

2.08716 

.50076 

1.99695 

.52279 

1.91282 

24 

37 

.45819 

2.18251 

.47948 

2.08560 

.50113 

1.99550 

.52316 

1.91147 

23 

38 

.45854 

2.18084 

.47984 

2.08405 

.50149 

1.99406 

.52353 

1.91012 

22 

39 

.45889 

2.17916 

.48019 

2.08250 

.50185 

1.99261 

.52390 

1.90876 

21 

40 

.45924 

2.17749 

.48055 

2.08094 

.50222 

1.99116 

.52427 

1.90741 

20 

41 

.45960 

2.17582 

.48091 

2.07939 

.50258 

1.98972 

.52464 

1.90607 

19 

42 

.45995 

2.17416 

.48127 

2.07785 

.50295 

1.98828 

.52501 

1.9047'2 

18 

43 

.46030 

2.17249 

.48163 

2.07630 

.50331 

1.98684 

.52538 

1.90337 

17 

44 

.46065 

2.17083 

.48198 

2.07476 

.50368 

1.98540 

.52575 

1.90203 

16 

45 

.46101 

2.16917 

.48234 

2.07321 

.50404 

1.98396 

.52613 

1.90069 

15 

46 

.46136 

2.167.51  1 

.48270 

2.07167 

.50441 

1.98253 

.52650 

1.899,35 

14 

47 

.46171 

2.16585  1 

.48306 

2.07014 

.50477 

1.98110 

.52687 

1.89801 

13 

48 

.46206 

2.16420  ! 

.48342 

2.06860 

.50514 

1.97966 

.52724 

1.89667 

12 

49 

.46242 

2.16255  i 

.48378 

2.06706 

.50550 

1.97823 

.52761 

1.89.533 

11 

50 

.46277 

2.16090  1 

.48414 

2.06553 

.50587 

1.97681 

.52798 

1.89400 

10 

61 

.46312 

2.1.5925 

.48450 

2.06400 

.50623 

1.97538 

..52836 

1.8926b 

9 

62 

.46348 

2.15760 

.48486 

2.06247 

.50660 

1.97395 

.52873 

1.89133 

8 

63 

.46383 

2.1.5.596 

.48521 

2.06094 

.50696 

1.97253 

.52910 

1.89000 

7 

64 

.46418 

2.154.32 

.48557 

2.05942 

.50733 

1.97111 

.,52947 

1.88867 

6 

55 

.464.54 

2.15268 

.48.593 

2.05790 

.50769 

1.96969 

.52985 

1.88734 

5 

56 

.46489 

2.15104 

.48629 

2.05637 

.50806 

1.96827 

.5,3022 

1.88602 

4 

67 

.46525 

2.14940 

.48665 

2.0.5485 

.50843 

1.96685 

.5,30.59 

1.88469 

3 

68 

.46560 

2.14777 

.48701 

2.0.53:« 

.50879 

1.96.544 

.53096 

1.88337 

2 

69 

.46595 

2.14614 

.48737 

2.05182 

..50916 

1.96402 

.53134 

1.88205 

1 

60 

.46631 

2.14451 

.48773 

2,0:50.30 

.50953 

1.96261 

.53171 

1.88073 

_0 

/ 

Cotang 

Tang 

Cotang  j 

Tang 

1 Cotang 

Tang 

Co  tang 

Tang 

/ 

65“ 

64“ 

1 63“ 

62“ 

464 


664 


SUR  VE  YJNG. 


TABLE  VII,  — Contiiincd. 
Natural  Tangents  and  Cotangents, 


28« 

! 29° 

0 

0 

CO 

31° 

Tariff 

Cotanff 

Tang 

Cotang 

Tang 

! Cotang 

_Tang 

1 Cofang 

/ 

0 

,53171 

1.88073 

.5.5431 

1.80405 

.677.35 

1.7:320.'r 

.GOOHC, 

1 .6(i42H' 

1 

.53208 

1.87941 

..5.5469 

1.80281 

..67774 

1.7:3089 

.60126 

; 1.6(i;31H 

59 

2 

.63246 

1.87-809 

..5.5.507 

1.801.58 

.67813 

1.72973 

.6016.-) 

1 1 rT)2(H) 

r)8 

3 

.6328:4 

1.87677 

..5.5.545 

1.80034 

.67851 

1.728.67 

.60205 

1 1 WKIOO 

57 

4 

.63320 

1.87546 

..5.5.5K3 

1.791)11 

,57890 

1.72741 

.00245 

1 1,6.5990 

.56 

5 

.53358 

1.87415 

..5.5621 

1.79788 

..67929 

1.72625 

.G02K1 

1 1.6.5K81 

55 

C 

.63:495 

1.87283 

.55659 

1 . 7 966.5 

..67968 

1.72509 

.60:321 

1.6.5772 

54 

7 

.5:4432 

1.87152 

..5.5697 

1.79.542 

..68(X)7 

1.72393 

.60.361 

1.6.566.3 

.53 

8 

.5:4470 

1.87021 

..557:10 

1.79419 

.58046 

1.72278 

.60103 

j 1 . 6.5.5.54 

52 

9 

.53507 

1.86891 

..5.5774 

1.79296 

. .6808.5 

1.72163 

.60113 

! 1.6.5-145 

51 

10 

.53545 

1.86760 

.55812 

1.79174  1 

.58124 

1.72047  1 

.60483 

1.65337 

50 

11 

..53.582 

1.86630 

..5.5a50 

1.79a51  1 

.58162 

1.710,32 

.60522 

1 1.6.5228 

49 

12 

..5:4620 

1.86499 

..5.588.S 

1.78929 

1 .58201 

1.71817 

.60.562 

1 1.65120 

48 

13 

..5.3657 

1.86369  , 

.5.5926 

1.78807  ' 

.58240 

1.71702  . 

, .60602 

1 1.6.5011 

47 

14 

..5.3694 

1.862.39 

..5.5964 

1.78685 

.58279 

1.71588  , 

1 .60642 

1 1.6.19tt3 

46 

15 

..5:4732 

1.86109 

..50003 

1.78503  1 

.68318 

1.71473 

1 .00681 

1 1.61795 

45 

16 

.53769 

1.85979 

..56041 

1.78441  i 

.58357 

1.7ia68 

1 .60721 

1 1.64687 

44 

17 

..5.3807 

1 . 85850 

..50079 

1.7a319 

.58396 

1.71244  ; 

1 .6076] 

1.64.579 

43 

18 

.53844 

1.85720 

..56117 

1.78198  , 

.58435 

1.71129 

i .60801 

1.64171 

42 

19 

.5.3882 

1.8.5591 

.501.56 

1.78077 

..68474 

1.71015 

.60841 

1.64.363 

41 

20 

.53920 

1.85462 

.50194 

1.77955  j 

.58513 

1.70901 

.66881 

1.04256 

40 

21 

.53957 

1.853.33 

..50232 

1.77a34 

: ..6,8552 

1.70787 

.60921 

1.64148 

39 

22 

.53995 

1.85204  1 

.50270 

1.77713 

..68591 

1.7007'3 

.609fX) 

1.64041 

38 

23 

.540:42 

1.85075 

.56309 

1.77592 

..686:31 

1 .70560 

.61000 

1.0.39.34 

37 

.54070 

1.84946 

.56347 

1.77471 

.58670 

1.70446 

.61040 

1.6.3826 

36 

25 

.54107 

1.84818 

.56.385 

1.77.3,51 

.58709 

1.70.3.32 

.61080 

1.6.3719 

35 

26 

.54145 

1.84689 

.56424 

1.77230 

.58748 

1.70219 

.61120 

1.63012 

34 

27 

.54183 

1.84561 

..50462 

1.77110 

.58787 

1.70106  1 

.61160 

1.6.3.505 

33 

28 

.54220 

1.84433 

.50501 

1.70990 

.58826 

1.C9992 

.61200 

1.0.3.398 

32 

29 

..54258 

1.84305 

.56639 

1.70869 

.58805 

1.69879 

.61240 

1.6.3292 

31 

30 

.54296 

1.84177 

.50577 

1.76749 

.58905 

1.69766 

.61280 

1.63185 

30 

31 

.54333 

1.84049 

.56616 

1.76629 

.58944 

1.69653 

.61.320 

1.63079 

29 

32 

.54371 

1.83922 

.56654 

1.76510 

.58983 

1.69541 

.61360 

1.62912 

28 

33 

.54409 

1.83794 

.50693 

1.76390 

.59022 

1.69428 

.61400 

1.62860 

27 

34 

.54446 

1.83667 

.56731 

1.76271 

.59061 

1.69316 

; .61440 

1.62760 

26 

35 

.54484 

1.83540 

.56769 

1.76151 

.59101 

1.69203 

.61480 

1 .62654 

25 

36 

.54522 

1.83413 

.56808 

1.76032 

.59140 

1.69091 

.61520 

1.62.548 

24 

37 

..54560 

1.83286 

.56846 

1.75913 

1 .591';9 

1.68979 

.61561 

1.62442 

23 

38 

.54597 

1.83159 

.56885 

1 75794 

.59218 

1.68866 

.61601 

1.62a36 

22 

39 

.54635 

1.83033 

.56923 

1.75675 

i .59258 

1.68754 

.61641 

1.622.30 

21 

40 

.54673 

1.82906 

^6962 

1.75556 

.59297 

1.68643 

.61681 

1.02125 

20 

41 

54711 

1.82780 

.57000 

1.75437 

.59336 

1.68531 

.61721 

1.62019 

19 

42 

.54748 

1.82654 

.57039 

1.75319 

.59376 

1.68419 

.61761 

1.01914 

18 

43 

.54786 

1.82528 

.57078 

1.75200 

.59415 

1.68308 

.61801 

1.61808 

17 

44 

.54824 

1.82402 

.57116 

1.75082 

.59454 

1.68196 

.61842 

1.61703 

16 

45 

.54862 

1.82276 

.57155 

1.74964 

.59494 

1.68085 

.61882 

1.61598 

15 

46 

.54900 

1.82150 

.57193 

1.74846 

.59533 

1.67974 

.61922 

1.61493 

14 

47 

.54938 

1.82025 

.57232 

1.74728 

.59573 

1.67863 

.61962 

1.61388 

13 

48 

.54975 

1.81899 

.57271 

1.74610 

.59612 

1.67752 

.62003 

1.61283 

12 

49 

.55013 

1.81774 

.57309 

1.74492 

.59651 

1.67641 

.62043 

1.61179 

11 

50 

.55051 

1.81649 

.57348 

1.74375 

1 .59691 

1.67530 

.62083 

1.61074 

10 

51 

.5.5089 

1.81.524 

.57386 

1.74257 

.59730 

1.67419 

.62124 

1.60970 

9 

52 

,.5.5127 

1.81399 

.,57425 

1.74140 

1 .59770 

1.67309 

.62164 

1.60865 

8 

53 

.55165 

1.81274 

..57404 

1.74022 

1 .59809 

1.67198 

.62204 

1.60761 

7 

54 

.55203 

1.81150 

.57503 

1.7.3905 

i .59849 

1.67088 

.62245 

1.60657 

6 

55 

.5.5241 

1.81025 

.57.541 

1.73788 

1 .59888 

1.66978 

.62285 

1.60553 

5 

56 

.5.5279 

1.80901 

.57,580 

1.7:3671 

.59928 

1.66867 

.62325 

1.60449 

4 

57 

.55317 

1.80777 

.57619 

1.7:3555 

.59967 

1.66757 

.62366 

1.60345 

3 

58 

.5.5.355 

1.806.53 

.576.57 

1.7:3438 

! .60007 

1.66647 

.62406 

1.60241 

2 

59 

.5.5393 

1.80.529 

.57696 

1.73321 

.60046 

1.66538 

.62446 

1.60137 

1 

60 

.5.5431 

1 .H0405 

..57735 

1.73205 

.60086 

1.66428 

.62487 

1.60033 

0 

Colang 

Tang 

Cotang  1 

Tang 

Cotang  1 

Tang 

Cotang  1 

Tang 

/ 

61» 

0 

0 

0 

1 69°  1 

68° 

TABLES. 


665 


TABLE  VII. — Continued. 


Natural  Tangents  and  Cotangents. 


32° 

33° 

CO 

35° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang  1 

Tang 

Cotang 

0 

.62487 

1.60033 

.64941 

1.53986 

.67451 

1.48256 

.70021 

1.42815 

1 

.62527 

1.59930 

.64982 

1.53888 

.67493 

1.48163  1 

.70064 

1.42726 

.62568 

1.59826 

.65024 

1.53791 

.67536 

1.48070 

.70107 

1.42638 

3 

.62608 

1.59723 

. 65065 

1.53693 

.67578 

1.47977 

.70151 

1 .42550 

4 

.62649 

1.59620 

.65106 

1.53595 

.67620 

1.47885 

.70194 

1.42462 

5 

.62689 

1.59517 

.65148 

1.53497 

.07603 

1.47792 

.70238 

1.42374 

6 

.62730 

1.59414 

.65189 

1.53400 

.67705 

1 .47699 

.70281 

1.42286 

7 

.62770 

1.59311 

.65231 

1.53302 

.67748 

1.47607 

.70.325 

1.42198 

8 

.62811 

1.59208 

.65272 

1.53205 

.67790 

1.47514 

.70368 

1.42110 

9 

.62852 

1.59105 

.65314 

1.53107 

.67832 

1.47422  I 

.70412 

1.42022 

10 

.62892 

1.59002 

.65355 

1.53010 

.67875 

1.47330 

.70455 

1.41934 

11 

.62933 

1.58900 

.65397 

1.52913  1 

.67917 

1.47238 

.70499 

1.41847 

12 

.62973 

1.58797 

.65438 

1.52816  ; 

.67960 

1.47146 

.70542 

1.41759 

13 

.63014 

1.58695 

.65480 

1.52719 

.68002 

1.47053 

.70586 

1.41672 

14 

.63055 

1.58593 

.65521 

1.52622  ' 

.68045 

1.46962 

.70629 

1.41584 

15 

.63095 

1.58490 

.05503 

1.52525 

.68088 

1.40870 

.70673 

1.41497 

16 

.63136 

1.58388 

.05604 

1.52429 

.68130 

1.46778 

.70717 

1.41409 

17 

.63177 

1.58286 

.05646 

1.52332 

.08173 

1.40686 

.70760 

1.41322 

18 

.63217 

1.58184 

.05688 

1.. 52235 

.68215 

1.40595 

.70804 

1.41235 

19 

.63258 

1.58083 

.65729 

1.52139 

.08258 

1.46503 

.70848 

1.41148 

20 

.63299 

1.57981 

.65771 

1.52043 

.68301 

1.46411 

.70891 

1.41061 

21 

.63340 

1.57879 

.65813 

1.51946 

.68343 

1.40320 

.70935 

1.40974 

22 

.63380 

1.57778 

.65854 

1.51850 

.68386 

1.40229 

.70979 

1.40887 

23 

.63421 

1.57676 

.65896 

1.51754 

.08429 

1.40137 

.71023 

1.40800 

24 

.63462 

1 .57575 

.65938 

1.51658 

.08471 

1.46046 

.71066 

1.40714 

25 

.63503 

1.57474 

.65980 

1.51502 

.08514 

jL . 4o9od 

.71110 

1.40627 

26 

.63544 

1.57372 

.06021 

1.51460 

.08557 

1.45864 

.71154 

1.40540 

27 

.63584 

1.57271 

.66063 

1.51370 

.08600 

1.45773 

.71198 

1.40454 

28 

.6362.5 

1.57170 

.60105 

1.51275 

.08642 

1.4.5682 

.71242 

1.40367 

29 

.63666 

1.57069 

.66147 

1.51179 

.68685 

1.45592 

.71285 

1.40281 

30 

.63707 

1.56969 

.66189 

1.51084 

.68728 

1.45501 

.71329 

1.40195 

31 

.63748 

1 .56868 

.66230 

1.50988 

.68771 

1 .45410 

.71373 

1.40109 

32 

.63789 

1.. 56767 

.66272 

1.50393 

.08814 

1.45320 

.71417 

1.40022 

33 

.63830 

1.56067 

.66314 

,1.50797 

.68857 

1.45229 

.71461 

1.39936 

34 

.63871 

1.56566 

.66350 

1.50702 

.63900 

1.45139 

.71505 

1.39850 

35 

.63912 

1.50466 

.66398 

1.50607 

.68942 

1.45049 

.71549 

1.39764 

36 

.63953 

1.. 50306 

.00440 

1.50512 

.08985 

1 .44958 

.71593 

1.39679 

37 

.63994 

1.. 56205 

.66482 

1.50417 

.69028 

1.44808 

.71637 

1.39593 

38 

.64035 

1.. 56105 

.66524 

1.50322 

.69071 

1.44778 

.71681 

1.39507 

39 

.64076 

1.56065 

.66566 

1.50228 

.69114 

1.4-1088 

.71725 

1.39421 

40 

.64117 

1.55966 

.66608 

1.50133 

.69157 

1.44598 

.71769 

1.39336 

41 

.641.58 

1.. 55806 

.66650 

1.50038 

.69200 

1.44.508 

.71813 

1.39250 

42 

.64199 

1 . 55766 

.66692 

1.49944 

.69243 

1.44418 

.71857 

1.39165 

43 

.64240 

1.55666 

.66734 

1.49849 

.69:230 

1.44329 

.71901 

1.39079 

44 

.64281 

1.55567 

.66776 

1 .49755 

.69329 

1.44239 

.71946 

1.38994 

45 

.64322 

1.55467 

.00818 

1.49661 

.09372 

1.44149 

.71990 

1.38909 

46 

.64363 

1.55.368 

.60860 

1.49506 

.09416 

1.44060 

.72034 

1.38324 

47 

.64404 

1.5.5269 

.66902 

1.49472 

.09459 

1.43970 

.72078 

1.38738 

48 

.64446 

1.5.5170 

.06944 

1.49378 

.09502 

1.43881 

.72122 

1.38653 

49 

.64487 

1.5.5071 

.66986 

1.49284 

.69545 

1.43792 

.72167 

1.38568 

50 

.64528 

1.54972 

.07028 

1.49190 

.69588 

1.43703 

.72211 

1.38484 

51 

.64.569 

1.54873 

.67071 

1.49097 

.69631 

1.43614 

.72255 

1.38399 

52 

.64610 

1.54774 

.67113 

1.49003 

.69675 

1.4:3525 

.72299 

1.38314 

53 

; .64652 

1., 54675 

; .07155 

1.48909 

.69718 

1.4:34:36 

.72344 

1.38229 

54 

: .64693 

1.54.576 

! .67197 

1.48816 

.69761 

1.4.3.347 

.72388 

1.38145 

55 

p .647:44 

1.. 54478 

.672:49 

1.48722 

! .69804 

1.432.58 

.72432 

1.38060 

5e 

i . 647’i  5 

1.. 54:479 

i .67282 

1.48629 

i .69847 

1.4.3169 

.72477 

1.37976 

51 

' .64817 

1.54281 

1 .67:424 

1.48536 

! .69891 

1.43080 

.72.521 

1.37891 

5f. 

! .648.58 

1.54183 

1 .67:466 

1.48442 

1 .69934 

1.42992 

.72565 

1.37807 

5i 

> .64899 

1.540a5 

‘ .67409 

1.48:449 

.69977 

1.42903 

.72610 

1.37722 

C( 

) .64941 

1.. 5:4986 

.67451 

1.48256 

' .70021 

1.42815 

.72654 

1.37638 

/ 

Cotang 

1 Tang 

Cotang 

1 Tang 

i Cotang 

Tang 

, Cotang 

Tang 

57° 

1 56° 

i 65° 

il  51° 

/ 

GO 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

43 

47 

40 

45 

44 

43 

42 

41 

40 

39 

33 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

28 

25 

24 

23 

22 

21 

20 

19 

10 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

_0 

_ / 


666 


SUR  VE  Y TNG. 


TABLE  VW.  — Continued. 
Natural  Tangents  and  Cotangents. 


36“ 

87° 

38° 

39° 

Tanpr J 

Cotiingf 

Taiih' 

C'otang 

Tang 

Cotang  j 

Tang 

< 'ot  arig 

0 

.72651 

1 .376;48 

.7.53.55 

1.. 32701  ; 

.78129 

Tr2799  r 

.H0!)78  1 

T.vsiTio 

60 

1 

.7269'.) 

1.37.554 

.7r)401 

1.32624 

.78175 

1.27917 

1 .81027 

1 21116 

.59 

2 

.72743 

1.37470 

.7.5147 

1.32514 

.78222 

1.27811 

.81075 

1.2:1:11.3 

.58 

3 

.72788 

1.37380 

.75192 

1.. 32164 

.78269 

1.27764  1 

.81123 

1.21270 

57 

4 

.72a32 

1 ..37.302 

.7.5.538 

1.323H4 

.78316 

1.27GS8  j 

.81171 

1.23196 

G6 

5 

.72877 

1.37218 

.7.5.58.4 

1.32304 

.78363 

1.27611 

.81220 

1.21123 

.55 

f) 

.72921 

1.. 371. 34 

.7.5629 

1.32224 

.78110 

1.27.535 

.81268 

1.2)0.50 

.54 

7 

.72966 

1.370.50 

.75675 

1.32144 

.784.57 

1.274.58 

.81316 

1.22977 

•53 

8 

.73010 

1 ..36967 

.7.5721 

1.32064 

.78504 

1.27;W2  1 

.813W 

1.22901 

52 

9 

.730.55 

1.368S3 

.75707 

1.319S4 

.78551 

1.27306 

.81113 

1.22831 

.51 

10 

.73100 

1.. 36800 

.75812 

1.31904 

.78598 

1.27230 

.81461 

1.22758 

.50 

11 

.73144 

1.. 3671 6 

.75R58 

1.31825 

.78045 

1.271.53 

.81.510 

1.22685 

49 

12 

.73189 

1.36633 

.7.5904 

1.31745 

.78692 

1.27077 

.81.5.58 

1.22612 

48 

13 

.73231 

1.. 36549 

.75950 

1.31606 

.7-8739 

1.27001 

.81606 

1.22539 

47 

14 

.73278 

1.36466 

.75996 

1.31586 

.78786 

1.26925 

.816.55 

1.22*167 

40 

15 

.73323 

1.. 36383 

.76012 

1.31507 

.78834 

1.26849 

.81703 

1.22394 

4.5 

IG 

.73368 

1.36.300 

.76088 

1.31427 

.78881 

1.26774 

.81752 

1.22321 

44 

17 

.73413 

1.36217 

.761.44 

1.31343 

.78923 

1.26698 

.81800 

1.22219 

4.3 

18 

.734.57 

1.361.34 

.76180 

1.31269  1 

.78075 

1.26622 

.81849 

1.22176 

42 

19 

.73502 

1.330.51 

.76226 

1.31190 

.79022 

1.26.546 

.81898 

1.22104 

41 

20 

.73547 

1.35'J63 

.76272 

1.31110 

.79070 

1.26471 

81946 

1.22031 

40 

21 

.73592 

1.3.5885 

.76318 

1.31031 

.79117 

1.26395 

.81995 

1.21959 

39 

22 

.73637 

1.S.5C02 

.76364 

1.30952 

.79164 

1.26319 

.82044 

1.21886 

;38 

23 

.73681 

1.35719 

.76410 

1.30073 

.79212 

1.26244 

.82092 

1.2I8I4 

37 

'iA 

.73726 

1.35637 

.764.56 

1.30795 

.79259 

1.26169 

.82141 

1.217-42 

36 

25 

.73771 

1.355.54 

.76502 

1.30716 

.79306 

1.26093 

.82190 

1.21670 

85 

2G 

.73816 

1.35472 

.76548 

1.30637 

.79354 

1. 2601 8 

.82238 

1.21598 

34 

27 

.73861 

1.35389 

.7-6594 

1 .30558 

.79401 

1.25943 

.82287 

1.21526 

.83 

£8 

.73906 

1.35307 

.70640 

i.saiBO 

.79449 

1.25867 

.82336 

1.214.54 

32 

29 

.73951 

1.35224 

.76686 

1.39401 

.79496 

1.2.5792 

.82385 

1.21382 

31 

30 

.73993 

1.35142 

.76733 

1.30323 

.79544 

1.25717 

.824.*^ 

1.21310 

30 

31 

.74041 

1.3.5060 

.76779 

1.30244 

.79591 

1.2.5642 

.82483 

1.21238 

20 

32 

.74033 

1.34978 

.76825 

1.30166 

.79639 

1.25567 

.82531 

1.21166 

23 

33 

.74131 

1.31896 

.76871 

1.30087 

.79036 

1.25492 

.82580 

1.21004 

27 

34 

.74176 

1.34814 

.76918 

1.30009 

.79734 

1.25-117 

.82629 

1.21023 

26 

35 

.74221 

1.34732 

.76964 

1.29931 

.79781 

1.25343 

.82678 

1.20951 

25 

36 

.74267 

1.34650 

.77010 

1.29653 

.79829 

1.25268 

.82727 

1.20879 

24 

37 

.74312 

1.34568 

.77057 

1.29775 

.79877 

1.25193 

.82776 

1.20808 

23 

38 

.74357 

1.34487 

.77103 

1.29696 

.79924 

1.25118 

.82825 

1.207.36 

22 

39 

.74402 

1.34405 

.77149 

1.29618 

.79972 

1.25044 

.82874 

1.20665 

21 

40 

.74447 

1.34323 

.77196 

1.29541 

.80020 

1.1^969 

.82923 

1.20593 

20 

41 

.74492 

1.34242 

.77242 

1.29463 

.80067 

1.24895 

.82972 

1.20522 

19 

42 

.74533 

1.34160 

.77289 

1.29385 

.80115 

1 .24820 

.83022 

1.204.51 

18 

43 

.74583 

1.34079 

.77-335 

1.29307 

.80103 

I.a4r46 

.83071 

1.20379 

17 

44 

.74628 

1.3.3998 

.77382 

1.29229 

.80211 

1.24672 

.83120 

1 .20308 

16 

45 

.74674 

1.33916 

.77423 

1.29152 

.80258 

1.24597 

.83169 

1 .20237 

15 

46 

.74719 

1.33835 

.77475 

1.29074 

.80306 

1.24523 

.83218 

1.20166 

14 

47 

.74764 

1.. 33754 

.77521 

1.28997 

.80354 

1.^49 

.63268 

1.20095 

13 

48 

.74810 

1.33673 

.77568 

1.28919 

.80402 

1.24375 

.63317 

1.20024 

12 

49 

.74855 

1.33.592 

.77615 

1.28842 

.80450 

1.24301 

.83366 

1.19953 

11 

50 

.74900 

l.:43511 

.77661 

1.287G4 

.80498 

1.24227 

.83415 

1.19882 

10 

51 

.74946 

1.334.30 

.77708 

1.28687 

.80546 

1.24153 

.83465 

1.19811 

9 

52 

.74991 

1.33349 

.77754 

1.28G10 

.80594 

1.24079 

.83514 

1.19740 

8 

53 

.75037 

1.33263 

.77801 

1.28533 

.80642 

1.24005 

.83564 

1.19669 

7 

54 

.75082 

1.33187 

.77848 

1.28456 

.80090 

1.23931 

.83613 

1.19599 

6 

55 

.75128 

1.. 33107 

.77895 

1.28379 

.80738 

1.23858 

.83662 

1.19528 

5 

56 

.75173 

1.3.3026 

.77941 

1.28302 

.80786 

1.23784 

.83712 

1.19457 

4 

57 

.75219 

1.. 32946 

.77988 

1.28225 

.80834 

1.23710 

.83761 

1.19387 

3 

58 

.75264 

1.32865 

.78035 

1.28148 

.80882 

1.23637 

.83811 

1.19316 

2 

59 

.75310 

1.32785 

.78082 

1.28071 

.80930 

1.2.3563 

.83860 

1.19246 

1 

GO 

.75355 

1.. 32704 

.78129 

1.27994 

.80978 

1.23490 

.83910 

1.19175 

0 

/ 

Cotaiif' 

1 Tang 

[Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

63° 

62° 

61° 

1 60° 

TABLES. 


667 


TABLE  Y\\.— Continued. 

Natural  Tangents  and  Cotangents. 


0 

41° 

42° 

43° 

Tang  1 Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang  1 

Cotang 

0 

.83910 

1.19175 

.86929 

1.15037 

.90040“ 

1.11061 

.932.52 

1.07237 

60 

1 

.83960 

1.19105 

.86980 

1.14969 

.90093 

1.10996 

.93306 

1.07174 

59 

2 

.84009 

1.19035 

.87031 

1.14902 

.90146 

1.10931 

.93360 

1.07112 

58 

3 

.84059 

1.18964 

.87082 

1.14834 

.90199 

1.10867 

.93415 

1.07049 

57 

4 

.84108 

1.18894 

.87133 

1.14767 

.90251 

1.10802 

.93469 

1.06987 

56 

5 

.84153 

1.18824 

.87184 

1.14699 

.90304 

1.10737 

.93524 

1.06925 

55 

6 

.84208 

1.187.54 

.87236 

1.14632 

.903.57 

1.10672 

.9.3578 

1.06862 

54 

7 

.84258 

1.18684 

.87287 

1.14565 

.90410 

1.10607 

.93633 

1.06800 

53 

8 

.81307 

1.18614 

.87338 

1.14498 

.90463 

1.10543 

.93688 

1.067.38 

52 

9 

.84357 

1.18544 

.87389 

1.14430 

.90516 

1.10478 

.93742 

1.06676 

51 

10 

.84407 

1.18474 

.87441 

1.14363 

.90569 

1.10414 

.93797 

1.06613 

50 

11 

.84457 

1.18404 

.87492 

1.14296 

.90621 

1.10349 

.93852 

1.06551 

49 

12 

.84507 

1.18334 

.87543 

1.14229 

.90674 

1.10285 

! .93906 

1.06489 

43 

13 

.84556 

1.18264 

.87595 

1.14162 

.90727 

1.10220 

1 .93961 

1.06427 

47 

14 

.84606 

1.18194 

.87646 

1.14095 

.90781 

1.101.56 

1 .94016 

1.06365 

46 

15 

.84656 

1.18125 

.67698 

1.14028 

.90834 

1.10C91 

i .94071 

1.06303 

45 

IG 

.84706 

1.18055 

.87749 

1.13961 

.90887 

1.10027 

.94125 

1.06241 

44 

17 

.84756 

1.17986 

.87801 

1.13894 

.90940 

1.00903 

.94180 

1.06179 

43 

18 

.&4806 

1.17916 

.87852 

1.13828 

.90993 

1. 09899 

.94235 

1.06117 

42 

19 

.84856 

1.17846 

.87904 

1.137G1 

.91046 

1 .09834 

.94290 

1.06056 

41 

20 

.84906 

1.17717 

.87955 

1.13694 

.91099 

1.09770 

.94345 

1.05994 

40 

21 

.84956 

1.17708 

.88007 

1.13627 

.91153 

1.09706 

.94400 

1.0,59.32 

39 

22 

.85006 

1.17638 

.88059 

1.13561 

.01206 

1.09642 

.944.55 

1.05870 

33 

23 

.85057 

1.17569 

.88110 

1.13494 

.91259 

1.09578 

.94510 

1.0,5809 

37 

24 

.85107 

1.17500 

.88162 

1.13423 

.91313 

1.09514 

.94565 

1.05747 

36 

25 

.85157 

1.17430 

.88214 

1.13361 

.91-366 

1.09450 

.94620 

1.05685 

35 

20 

.85207 

1.17361 

.83265 

1.13295 

.91419 

1.09386 

.94676 

1.05624 

34 

27 

.85257 

1.17292 

.88317 

1.13223 

.91473 

1.09322 

.94731 

1.0.5.562 

33 

28 

.85308 

1.17223 

.88369 

1.13162 

.91526 

1.09258 

.94786 

1.05501 

32 

29 

.85358 

1.17154 

.88421 

1.1 3096 

.91580 

1.09195 

.94841 

1.054.39 

31 

30 

.85408 

1.17085 

.8847'3 

1.13029 

.91633 

1.09131 

.94896 

1.05378 

30 

31 

.85458 

1.17016 

.88524 

1.12963 

.91687 

1.C90C7 

.94952 

1.05317 

29 

32 

.85509 

1.16947 

.88576 

1.12897 

.917-40 

1.C9003 

.95007 

1.05255 

28 

33 

.85559 

1.16878 

.88628 

1.12831 

.91794 

1.08940 

.95062 

1.05194 

27 

31 

.85609 

1.16209 

.88680 

1.12765 

.91847 

1.0887'6 

.9.5113 

1.0.5133 

26 

35 

.85660 

1.16741 

.88732 

1.12699 

.91901 

1.08813 

.95173 

1.05072 

25 

3G 

.85710 

1.16672 

.88784 

1.12633 

.91955 

1.08749 

.9.5229 

1.0.5010 

24 

37 

.85761 

1.16603 

.88836 

1.12567 

.92008 

1.08686 

.9.5284 

1.04949 

23 

38 

.85811 

1.16535 

.88888 

1.12501 

.92162 

1.08622 

.9.5340 

1.04888 

22 

39 

.a58C2 

1.1G4C6 

.88940 

1.12435 

.92116 

1.03559 

.9.5.395 

1.04827 

21 

40 

.85912 

1.16398 

.68992 

1.12369 

.92170 

1.08496 

.95451 

1.04766 

20 

41 

.85963 

1.16329 

.89045 

1.12303 

.92224 

1.08432 

.95.506 

1.04705 

19 

42 

.86014 

1.1 6261 

.89097 

1.12238 

.92277 

1,08369 

.95.562 

1.04644 

18 

43 

.86064 

1.16192 

.89149 

1.12172 

.92331 

1.08306 

.95618 

1.04583 

17 

44 

.86115 

1.16124 

.89201 

1.12106 

.92385 

1.08243 

.9.5673 

1.04.522 

16 

45 

.86166 

1.16056 

.89253 

/I. 12041 

.92439 

1.0817'9 

.95729 

1.04461 

15 

4G 

.8621« 

1.15987 

.89306 

1.1 1975 

.92493 

1.08116 

.9.57-85 

1.04401 

14 

47 

.86267 

1.15919 

.89358 

1.11909 

.92:547 

1.08053 

.9.5841 

1.04.340 

13 

48 

.86318 

1.15851 

.89410 

1.11844 

.92601 

1 .07990 

.95897 

1.04279 

12 

49 

.86368 

1.15783 

.89463 

1 11778 

.92655 

1.07927 

.9.5952 

1.04218 

11 

50 

.86419 

1.15715 

.89515 

1.11713 

.92709 

1.07864 

.96008 

1.04158 

10 

51 

.86470 

1.15647 

.89567 

1.11648 

.92763 

1.07801 

.96064 

1.04097 

9 

52 

.86521 

1 . 15579 

.89620 

1.11.582 

.9»j817 

1.07738 

.96120 

1.04036 

8 

53 

.86.572 

1.15511 

.89672 

1.11.517 

.92872 

1.07676 

.96176 

1.03976 

7 

r>4 

.86623 

1.15443 

.89725 

1.11452 

.92926 

1.07613 

.962.32 

1.03915 

6 

55 

.86674 

1.15375 

.89777 

1.11387 

.92980 

1.07,5.50 

.96288 

1.03855 

5 

5G 

.86725 

1.1.5308 

.89830 

1.11321 

.9:4034 

1. 07487 

.96.344 

1.03794 

4 

57 

.86776 

1.1.5240 

.89883 

1.112.56 

.93088 

1.07425 

.96400 

1.0.3734 

8 

58 

.86827 

1.15172 

.899^5 

1.11191 

.93143 

1.07362 

.964.57 

1.03674 

2 

59 

.86878 

1.15104 

.89988 

1.11126 

.93197 

1.07299 

.96.513 

1.0.3613 

1 

GO 

.86929 

1.1.5037 

.‘K)040 

1.11061 

.932,52 

1.07237 

.96.569 

1 03.5.53 

0 

/ 

Cotang 

i Tang 

j Cotang  I Tang 

Cotang 

Tang 

Cotang 

'I'ang 

49“ 

0 

CO 

i 47°  ! 

i 46° 

668 


SURVEYING. 


TABLE  VII. — Continued. 

Natural  Tangents  and  Cotangents. 


9 

440 

. 

|, 

440 

/ 1 

1! . 

440 

/ 

Tang 

Cotang 

1 

Tang 

Cotang 

Tang 

Cotang 

0 

.96569 

1.03.5.53 

60 

20 

.97700 

1.02855 

40  ' 

!40 

.98843 

1.01170 

20 

1 

.96625 

1.0:3493 

59 

21 

.977.56 

1.02295 

1 '^>9  1 

1 41 

.98901 

1 01112 

19 

2 

.96681 

1.034.33 

58 

22 

.97813 

1.022:56  1 

.381 

42 

.089.58 

1 010.53 

18 

3 

.96738 

1.0;3172 

57 

23 

,97870 

1.02176  1 

37  i 

4.3 

.99016 

1.00994 

17 

4 

.96791 

1.0:3312 

56 

24 

.97927 

1.02117  ; 

36 : 

44 

.9(K)7.3 

1.009:35 

16 

5 

.90a50 

l.a32.52 

55 

25 

.9798'4 

1.020.57  1 

.85! 

45 

.99131 

1.00876 

15 

6 

.96907 

1.03192 

54 

26 

.98041 

1.011H)8 

.34 

! 46 

.99180 

1.00818 

14 

7 

.96963 

1.031.32 

53 

27 

.98098 

1.019.30 

:33 

47 

.99217 

1.00750 

13 

8 

.97020 

1 .0:1072 

52 

28 

.981.55 

1.01879 

32 

48 

.99.304 

1.00701 

12 

9 

.97076 

1.03012 

51 

29 

.98213 

1.01820 

31 

49 

.99.362 

1.00642 

11 

10 

.97133 

1.02952 

50 

:30 

.98270 

1.01761 

30 

50 

.99120 

1.00583 

10 

11 

.97189 

1.02802 

40 

31 

.08327 

1.01702 

20 

51 

.90178 

1.00.525 

9 

12 

.97246 

1.028.32 

48 

32 

.93384 

1.01612  , 

281 

52 

.99.536 

1.00467 

8 

13 

.97302 

1 .02772 

47 

.33 

.98441 

1.01.583  I 

27  1 

.5:3 

.99.594 

1.00408 

7 

14 

.97359 

1.02713 

46 

:34 

.08190 

1.01.521  ' 

26  1 

51 

.996.52 

1.00.350 

6 

15 

.97416 

1.026.53 

45 

.35 

.9,85.56 

1.01465 

25  > 

55 

.99710 

1.00291 

5 

16 

.97472 

1.02.503 

44 

.36 

.98613 

1.01406 

24 

.56 

.997(W 

1.00*233 

4 

17 

.97529 

1.02533 

43 

37 

.08671 

1.01.347  1 

23! 

' 57 

.998*26 

1.00175 

3 

18 

.97586 

1.02474 

42 

.38 

.98728 

1.01288 

22  ' 

58 

.09884 

1.00116 

2 

19 

.97643 

1.02414 

41 

.30 

.98786 

1.01*220 

21 

50 

.9994*2 

1.00058 

1 ! 

20 

.97700 

1.02:355 

40 

40 

.98843 

1.01170 

20' 

60 

1 .00000  1 

l.OOOOO 

0 1 

Cotang  1 

Tang 

/ 

/ 

Cotang 

Tang  1 

/ 

Cotang  1 

Tang 

45° 

45°  1 

45° 

= Dumber  degrees  of  longitude  between  the  given  meridian  and  the  prime  meridian  of  the  map. 


TABLES. 


669 


Co-ordinates  of  Points  of  Intersection  of  Parallels  and  Meridians  in  Polyconic  Projection.  § 417. 


Giving  Values  of  C in  Kutter’s  Formula  when  j‘  = p.ooi.  § 259. 


670 


SURVEYING. 


c 

^ « CO  ^ >5 

C X © © 

e>  -t « X © 

e»  -?  © X © 

-f  X ?>  c © 

^ ^ ^ ^ 

et  it  et  n 

X r.  t -I-  o' 

ro  in  00  0 

0 lO  « m 

fx  in  N VO  Os 

000  0"0 

0 m tx  tx  tx 

CO 

© 

SO  M 10  0 

M Ct  N ^ m 

n m m'O 
m m m m CO 

^ M 

txoo  O'  0 0 

^ sr  ^ m 

c«  m t m'O 
m m m m m 

0 

M ro  r*-i  M 

m m 0 -t  Os 

M W O'©  0 

w m c* 

00  CO  m rx  tx 

CO 

0 

0 VO  0 

N Cl  fO  m ro 

00  0 (s  m -t 
m ^ 

tx  0 •-  c<  t 
^ ^ m m m 

m'O  cxco  O' 
m m m m m 

n r«  rs*  in 
sc  VO  so  VO  vo 

iio 

C"0  ^ 

0 N c*  00 

M M m 0 m 

O'  « fi  m m 

0 moo  0 M 

© 

in  moo  n m 
w CO  CO  ^ 

00  0 c«  m 1/. 
^ m m m m 

00  0 w -t  m 
mvo  VO  VO  VO 

soooao- 

so  VO  so  tx  Cx 

m t m txoo 

tx  tx  tx  tx  tx 

IS 

e» 

e? 

c 

0 'O  so  -^00 

m O'  0 00  m 

m cv  0"0  -t 

|X  0 •-'  w « 

0 m Os  « m 

0 00  -too 

CO  ro  '•t  Tf  so 

f SO  0 0 
m m mso  vo 

m O'  cn 

so  VO  VO 

«tvO  txOO  Os 
rx  Cx  fx  rx  Cx 

•-  ri  m mvo 

00  CO  00  00  GO 

© 

in  N 00  N 0 

0 -t  m 

m 0 m ft  Os 

m.so  0 « m 

w m 't-'O 

tt 

0 

»o  in  SO  Q 
CO  m mo 

\0 

^ tx.  Ov  « n 
tx  tx.  txOO  00 

-t  m rxoo  O' 
CO  CO  CO  CO  CO 

O'  O'  O'  Os'os 

so  W '«t00  00 

^ m O'  N 0 

Cl  m'O  ''t 

0 'too  0 

M CO  mvo  Os 

0 

•^vo  moo 
mvo  VO 

VO  O'  '^'O 

1^00  00  00 

<»  O'  O'^'os 

Q >-•  Cl  -t  m 
00000 

^<3  3:2 

b, 

0 

in 

Id 

D 

.J 

< 

> 

I 

.015 

N CO  0 

mio  m 0 m 
m'O  c^oo  00 

00  0 VO  0 M 

00  N -t  c^  Os 
00  Os  O'  Os  Os 

tx  cx  w m.  m 

f»  moo  0 N 

0 0 0 « -i 

0 -too  0 Cl 

t m-o  CO  O' 

m 0 VO  O'  ci 

M m m tx 

W d M Cl  W 

CO 

0 N (N  W N 

m t^vo  w m 

tx  c<  tx  m 

m O'  CO  IX  O' 

0 CO  -stco  0 

c 

m 0 0 VO  *-• 

VO  CO  Os  O'  0 

moo  ^ ’^t'O 

0 0 « « M 

0 mvo  00  0 
c<  cs  c<  c*  m 

Cl  m m'O  tx 
m rr.  m m,  m 

0 HI  ro  -^vo 
-t  -St  -t  -^r 

tv  « 00  0 w 

m m M 

m cx.  ^ ^xoo 

tx  m.oo  M m, 

-t  m 0 tx 

H 

0 ; 

W C'OO  VO  « 

r^oo  O'  0 « 

m O'  c<  m 

M »-i  CJ  N 

M xj-  fx  O'  >-• 
m m m m 

m\0  00  O' 

M m.  mvo  tx 
m m m,  u-i  m 

N 0 0 0 00 

m O'  M 00  m 

VO  OsCO  W 

m.  0 moo  N 

CO  Cl  00  mso 

r=l 

© 

c<  0 •-  00  m 
00  0 M M o» 

00  ►-  m hx  0 
ci  m m m ''I- 

'St  cx  0 m m 
xt  m m m 

tx  O'  0 •-  m 
m m\c  VO  VO 

m txoo  0 

VC  VO  VO  IX  tx 

0 

CO  M 00  mvo 

m ''too  w 

•♦00  m'O 

m Cl  CO  w ^ 

txvo  moo  M 

0 

m.  m m N 00 
Ov  ^ C4  m m 

m C'x  0 co*o 
•'T  ''i*  m m m 

0 't  Cx  O'  M 
SO  VO  VO  VO  Cx 

m m.vo  CO  O' 

tx  tx  tx  tx  rx 

M m mvo  CO 

cc  00  CO  00  00 

0 

m moo  moo 

O'  w C^OO  "t 

0 VO  tx  N 'St 

mo  t-  0 rn 

tx  tx  '.t  0 m 

© 

© 

00  O'  0 VO 

0 cs  m m 

M VO  O'  01  m 
so  VO  VO 

0 mvo  O'  ^ 
CO  00  00  00  Os 

m.  mvo  00  O' 
O'  O'  O'  O'  O' 

M m m txoo 

00000 

Cl  Cl  Cl  Cl  Cl 

r in 
feet. 

H N CS  Tjl  LS 

©t'X©© 

N^OX© 

HHHHM 

N'i'XX© 

«W««N 

'!j‘  X e?  © © . 

p:  ec  ^ ^ ‘-'5 

TABLE  X.  § 25q. 

Giving  Diameters  in  Feet  of  Circular  Brick  Conduits  for  Various  Inclinations  and  Rates  of  Discharge. 

Conduit  full  to  point  of  maximum  discharge.  (By  Kutter’s  formula.)  § 239. 


TABLES. 


671 


0) 

•a 
- a 

H«CO'i<iO©W5©»0©LO©»f5©©©©©©K50»0©©®®0©®©00© 

0 

LI  0 

HHeiNCOCO'^'<#»O©^-0O®®WLOi^®LO®»O®®®®®®® 

3 c/5 

r-ltHr-lHCiWCOCOTi^lOOi'^QOOO 

U 

0 

'^lO  ^ ^ ^ ^ *^00  O'  0 W w ’^'O  1^00  M comtxO  roioi^O  N 

d 

2 

2 

2 

2 

2 

2 

2 

3 

3 

3 

3 

4 

4 

4 

4 

5 

5 

H 

© 

vnvo  00  0 ro  iii  invo  t^oo  o O m n ro  ti-vo  oo  0 m OO  0 roiot^ 

d 

0 

irivo  t^oo  OOO  W lovo  00  O Cl  -.j-t^OCl  T^t^H  -^t^OvN 

rHHHHH  ^ rorofnroro-.j-T)-rj-ir)ioic)  lo'O 

0 

ir,  t^oo  COgJfj^ifljsoci®®®  ” co-'finc^t^O  Cl  '4-'0  O'  m -.^-O  0 C^  0 Cl  uo 

eo 

0 

VO  C'oo  O'Of0»r5®X®®HC'te0vaOi^  OiO  n 'i-vo  oo  « m o -vl-oo  M in  0 

N 

H 

VO  C'  O'  0 0 r«C50i.^fffM^Cffl?'.(Ji^«M»^vooo  O •v^c^0  Cl  t^w  looo  I-I 

u 

w 

H 

” ” ” H H H « N W « Ci  N « « CO  c?  CO  'C-  •^'O  'O  VO  c^ 

0 

0 

0 

VO  00  c>0  » h °. 

H 

” ” ” ” ” N N « N N N N N CO  CO  CO  CO  CO  rji 

U 

t^oo  0 •-  M moo  0 « 'CiO  CD  t'*  ® H CO  rj<  ® ® H O ® Tji  0 mo  'd-oo  m m 

< 

„ „ M M M M Cl  N ej;'  {*}■  ((j'  e,’  e^'  {,'  ^ ^ 1010'°'®  C'  C'  00 

0 

O'  M N mvo  Ov  M m mvo  r^ovo  Pc^lC?'*Xeii.O®®^OOHl.O®»-0®  c^ 

HMMHiHcicicicicicim  roj^  CO  M M "ji  ^ LO  LO  UO  0 CD  r-  r*  X °°  °°  ^ 

c 

00  0 Cl  m ^co  M mmc^OvO  w mmi^o  w dcecjNvjjcs^Qo^acMXCOr^H 

CO 

„ „ „ „ „ d Cl  Cl  Cl  Cl 

H 

© 

00  H m■'^mo^mmt^ovM  ci  r^m^^o  « r^vo  o mvo  05Tf(®«r’^®»0®U5Ci 

N 

H M M M M d d d d 

1H  HH 

10 

O'  M m mvo  0 -vJ-vo  O'  M m -vj-vo  c^  o ci  -.i-vo  oo  d vo  O'  moo  Xf^rHX»0®®iHia 

H 

mmhhwWNN  ^ 

rH  H H H 

0 

Os  c*  Th\o  t^NVO  Ow  fOiot^OO  ro  irioo  O Cl  m ^co  ro  o*  moo  IC  O'?  X ^ Ci  ^ 

iM  H M M N w 0 lovo  'O  \o  t^oo  oo  ^ q ^ ^ ^ 

rH  H iH  H 

flj 

*0 
u a 

HNMTj(>a®»D®».'5®>'i®»D®®®®®®W®»«®®®®®®0®©®© 

0 

V 0 

r-IH0JNXM^^»0©i''XO®«l0?'®»0O»D®®®®®®® 

!5 

CL  0 

r1i-ltHHW0lMMT)(ID®r-X®® 

3 

u 

t/) 

H 

672 


S UR  VE  YING. 


TAIU.E  XI. 


Volumes  by  the  Prism'jid/l  Formula.  § 320. 


Widths. 

Heights. 

Corrections 
for  ti  nths 
in  height. 

1 

2 

3 

5 

6 

7 

8 

9 

10 

1 

0 

1 

1 

1 

2 

0 

2 

2 

3 

3 

. I 

0 

2 

1 

1 

0 

2 

3 

3 

4 

5 

6 

6 

.2 

0 

» 

1 

2 

3 

4 

5 

6 

6 

7 

8 

9 

.3 

0 

4 

1 

2 

4 

5 

6 

7 

9 

10 

11 

12 

.4 

1 

6 

2 

—3 

— 5 

-6 

—8 

—9 

—11 

—12 

—14 

—15 

.5 

1 

6 

2 

4 

6 

7 

9 

11 

13 

15 

17 

19 

.6 

1 

7 

2 

4 

G 

9 

11 

13 

15 

17 

19 

22 

.7 

1 

8 

2 

5 

7 

10 

12 

15 

17 

20 

22 

25 

.8 

1 

{) 

3 

0 

8 

11 

14 

17 

19 

22 

25 

28 

•9 

1 

10 

3 

0 

9 

12 

15 

19 

22 

25 

28 

31 

11 

3 

7 

10 

14 

17 

20 

24 

27 

31 

34 

.1 

0 

12 

4 

7 

11 

15 

19 

22 

26 

30 

:« 

37 

.2 

1 

13 

4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

.3 

1 

14 

4 

9 

13 

17 

22 

26 

30 

35 

39 

43 

• 4 

2 

16 

—5 

—9 

—14 

—19 

—23 

—28 

—32 

— 37 

—42 

—46 

• 5 

2 

10 

6 

10 

15 

20 

25 

30 

35 

40 

44 

49 

.6 

3 

17 

5 

10 

16 

21 

26 

31 

37 

42 

47 

52 

.7 

3 

18 

6 

11 

17 

22 

28 

33 

39 

44 

50 

56 

.8 

4 

19 

6 

12 

18 

23 

29 

;i5 

41 

47 

53 

59 

•9 

4 

20 

6 

12 

19 

25 

31 

37 

43 

49 

56 

62 

21 

6 

13 

19 

26 

32 

39 

45 

52 

58 

65 

, t 

1 

22 

7 

14 

20 

27 

34 

41 

48 

54 

61 

68 

.2 

2 

23 

7 

14 

21 

28 

35 

43 

50 

57 

64 

71 

.3 

2 

24 

7 

15 

22 

30 

37 

44 

52 

59 

67 

74 

.4 

3 

25 

—8 

— 15 

—23 

-31 

—39 

—46 

—54 

—62 

—69 

—77 

.5 

4 

26 

8 

16 

24 

32 

40 

48 

56 

64 

72 

80 

.6 

5 

27 

8 

17 

25 

33 

42 

50 

58 

67 

75 

83 

.7 

5 

28 

9 

17 

26 

35 

43 

52 

60 

69 

78 

86 

.8 

6 

29 

9 

18 

27 

36 

45 

54 

63 

72 

81 

90 

•9 

7 

80 

9 

19 

28 

37 

46 

56 

65 

74 

83 

93 

81 

10 

19 

29 

38 

48 

57 

67 

77 

86 

96 

.1 

1 

82 

10 

20 

30 

40 

49 

59 

69 

79 

89 

99 

.2 

2 

83 

10 

20 

31 

41 

51 

61 

71 

81 

92 

102 

.3 

3 

84 

10 

21 

31 

42 

52 

63 

73 

84 

94 

105 

•4 

4 

35 

—11 

-22 

—32 

—43 

—54 

—65 

—76 

—86 

-97 

—108 

• 5 

5 

36 

11 

22 

33 

44 

56 

67 

78 

89 

100 

111 

.6 

6 

37 

11 

23 

34 

46 

57 

69 

80 

91 

103 

114 

.7 

8 

38 

12 

23 

35 

47 

59 

70 

82 

94 

106 

117 

.8 

9 

39 

12 

24 

36 

48 

60 

72 

84 

96 

108 

120 

•9 

10 

40 

12 

25 

37 

49 

62 

74 

86 

99 

111 

123 

41 

13 

25 

38 

51 

63 

76 

89 

101 

114 

127 

.1 

1 

42 

13 

26 

39 

52 

65 

78 

91 

104 

117 

130 

.2 

3 

43 

13 

27 

40 

53 

66 

80 

93 

106 

119 

133 

.3 

4 

44 

14 

27 

41 

54 

68 

81 

95 

109 

122 

136 

•4 

6 

45 

—14 

-28 

—42 

—56 

—69 

-83 

—97 

-111 

—125 

—139 

.5 

7 

46 

14 

28 

43 

57 

71 

85 

99 

114 

128 

142 

.6 

8 

47 

15 

29 

44 

58 

73 

87 

102 

116 

131 

145 

.7 

10 

48 

15 

30 

44 

59 

74 

89 

104 

119 

133 

148 

.8 

11 

49 

15 

30 

45 

60 

76 

91 

106 

121 

136 

151 

.9 

13 

60 

15 

31 

46 

62 

77 

93 

108 

123 

139 

154 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

. I 

.2 

•3 

•4 

■ 5 

.6 

•7 

.8 

•9 

Corrections  for 
tenths  in  width. 

0 

0 

0 

1 

1 

1 

1 

1 

TABLES. 


673 


TABLE  XI. — Contimied. 

Volumes  by  the  Prismoidal  Formula. 


Heights. 

Corrections 

1 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

in  height. 

61 

16 

31 

47 

63 

79 

94 

110 

126 

142 

1.57 

.1 

2 

62 

16 

32 

48 

64 

80 

96 

112 

128 

144 

160 

.2 

3 

63 

16 

33 

49 

65 

82 

98 

115 

131 

147 

163 

• 3 

5 

64 

17 

33 

50 

67 

83 

100 

117 

133 

1.50 

167 

•4 

7 

65 

—17 

-31 

—51 

—68 

—85 

—102 

—119 

—1.36 

—153 

—170 

.5 

8 

66 

17 

35 

52 

69 

86 

104 

121 

138 

156 

173 

.6 

10 

67 

18 

35 

53 

70 

88 

106 

123 

141 

158 

176 

.7 

12 

68 

18 

36 

54 

72 

90 

107 

125 

143 

161 

179 

.8 

14 

69 

18 

36 

55 

73 

91 

109 

127 

146 

164 

182 

•9 

15 

60 

19 

37 

56 

74 

93 

111 

130 

148 

167 

185 

61 

19 

38 

56 

75 

94 

113 

1.32 

151 

169 

188 

.1 

2 

62 

19 

38 

57 

77 

96 

115 

134 

153 

172 

191 

.2 

4 

63 

19 

39 

58 

78 

97 

117 

136 

156 

175 

194 

.3 

6 

64 

20 

40 

59 

79 

99 

119 

138 

158 

178 

197 

■4 

8 

65 

—20 

-40 

—60 

—80 

—100 

—120 

—140 

—160 

—181 

-201 

.5 

10 

66 

20 

41 

61 

81 

102 

122 

143 

163 

183 

204 

.6 

12 

67 

21 

41 

62 

83 

103 

124 

145 

165 

186 

207 

.7 

14 

68 

21 

42 

63 

84 

105 

126 

147 

168 

189 

210 

.8 

16 

69 

21 

43 

64 

85 

106 

128 

149 

170 

192 

213 

•9 

18 

70 

22 

43 

65 

86 

108 

130 

1.51 

173 

194 

216 

71 

23 

44 

66 

88 

100 

131 

153 

175 

197 

219 

.1 

2 

72 

22 

44 

67 

89 

111 

133 

156 

178 

200 

222 

.2 

5 

73 

23 

45 

68 

90 

113 

135 

1.58 

180 

203 

225 

.3 

7 

74 

23 

46 

69 

91 

114 

137 

160 

183 

206 

228 

•4 

9 

75 

—23 

—46 

—69 

—93 

-116 

—139 

—162 

— 185 

—208 

—231 

.5 

12 

76 

23 

47 

70 

94 

117 

141 

164 

188 

211 

235 

.6 

14 

77 

24 

48 

71 

95 

119 

143 

166 

190 

214 

238 

.7 

16 

78 

24 

48 

72 

96 

120 

144 

169 

193 

217 

241 

.8 

19 

79 

24 

49 

73 

98 

122 

146 

171 

195 

219 

244 

•9 

21 

SO 

25 

49 

74 

99 

123 

148 

173 

198 

222 

247 

81 

25 

50 

75 

100 

125 

1.50 

175 

200 

225 

250 

3 

82 

25 

51 

76 

101 

127 

152 

177 

202 

228 

253 

.2 

5 

83 

26 

51 

77 

102 

128 

154 

179 

205 

231 

256 

.3 

8 

84 

26 

52 

78 

104 

130 

156 

181 

207 

233 

259 

•4 

10 

85 

—26 

—52 

—79 

—105 

—131 

—1.57 

-184 

—210 

—236 

—262 

.5 

13 

86 

27 

53 

80 

106 

133 

159 

186 

212 

239 

265 

.6 

16 

87 

27 

54 

81 

107 

134 

161 

188 

215 

242 

269 

.7 

18 

88 

27 

54 

81 

109 

136 

163 

190 

217 

244 

272 

.8 

21 

89 

27 

55 

82 

110 

137 

165 

192 

220 

247 

275 

•9 

24 

90 

28 

56 

83 

111 

139 

167 

194 

222 

250 

278 

91 

28 

56 

84 

112 

140 

169 

197 

225 

253 

281 

.1 

3 

92 

28 

57 

85 

114 

142 

170 

199 

227 

256 

284 

.2 

6 

93 

29 

57 

86 

115 

144 

172 

201 

230 

258 

287 

.3 

9 

94 

29 

58 

87 

116 

145 

174 

203 

232 

261 

290 

•4 

12 

95 

-29 

—59 

-88 

—117 

—147 

-176 

—205 

—235 

—264 

—293 

.5 

15 

90 

30 

59 

89 

119 

148 

178 

207 

2:37 

267 

296 

.6 

18 

97 

30 

60 

90 

120 

150 

180 

210 

240 

269 

299 

.7 

21 

98 

30 

60 

91 

121 

151 

181 

212 

242 

272 

302 

.8 

23 

99 

31 

61 

92 

122 

153 

183 

214 

244 

275 

306 

•9 

26 

100 

31 

62 

93 

123 

151 

185 

216 

247 

278 

309 

1 

2 

i 3 

4 

5 

6 

7 

8 

1 9 

10 

. I 

.2 

1“ 

•4 

•5 

.6 

•7 

.8 

•9 

/ 

0 

0 

1 “ 

1 

1 

1 

1 

1 

1 

tenths  in  width. 

674 


SURVEYING. 


TAHLE  XI. — ContiftiicJ. 

Volumes  ijy  the  Prismoidal  Formula. 


(/) 

x: 

Heights. 

Correction* 
for  tenths 

11 

12 

1,3 

11 

15 

10 

' 1 

' i 

18 

19 

20 

in  height. 

1 

3 

4 

4 

4 

5 

5 

5 

0 

0 

0 

.T 

0 

a 

7 

7 

8 

9 

9 

10 

10 

11 

12 

12 

.3 

0 

3 

10 

11 

12 

13 

14 

15 

10 

17 

18 

19 

.3 

0 

4 

1 1 

15 

10 

17 

19 

20 

21 

22 

23 

25 

.4 

1 

r> 

—17 

—10 

—20 

00 

—23 

-25 

—20 

—28 

—29 

-31 

.5 

1 

(> 

20 

oo 

21 

20 

28 

30 

31 

33 

35 

37 

.6 

1 

1 

24 

20 

28 

30 

32 

.3.5 

37 

39 

41 

43 

.7 

1 

8 

27 

30 

32 

35 

37 

40 

42 

44 

47 

49 

.8 

1 

1) 

31 

33 

36 

39 

42 

44 

47 

.50 

.53 

56 

•9 

1 

10 

34 

37 

40 

43 

40 

49 

52 

50 

59 

02 

11 

37 

41 

44 

48 

51 

54 

58 

01 

05 

08 

0 

12 

41 

44 

48 

52 

50 

59 

63 

07 

70 

74 

.2 

1 

13 

44 

48 

52 

50 

on 

04 

08 

72 

70 

80 

.3 

1 

14 

48 

52 

50 

00 

05 

09 

73 

78 

82 

80 

•4 

2 

15 

—51 

—50 

—00 

—05 

—09 

—74 

—79 

—83 

—88 

—93 

.5 

2 

10 

54 

59 

04 

09 

74 

79 

84 

89 

94 

99 

.6 

3 

17 

58 

03 

08 

73 

79 

84 

89 

94 

100 

105 

.7 

3 

18 

01 

07 

70 

78 

8.3 

89 

94 

100 

106 

in 

.8 

4 

10 

05 

70 

70 

82 

88 

94 

100 

106 

111 

117 

•9 

4 

20 

08 

74 

80 

80 

93 

99 

105 

111 

117 

12.3 

21 

71 

78 

84 

91 

97 

104 

110 

117 

123 

130 

.1 

1 

22 

75 

81 

88 

95 

102 

109 

115 

122 

129 

1.36 

.2 

0 

23 

78 

85 

92 

99 

106 

114 

121 

128 

13.5 

142 

.3 

2 

24 

81 

89 

90 

104 

111 

119 

120 

133 

141 

148 

•4 

3 

25 

-85 

-93 

—100 

—108 

—116 

—123 

131 

-139 

-147 

—1.54 

.5 

4 

20 

88 

96 

104 

112 

120 

128 

130 

144 

152 

100 

.6 

5 

27 

92 

100 

108 

117 

125 

133 

142 

1.50 

1.58 

107 

.7 

5 

28 

95 

104 

112 

121 

130 

1.38 

147 

1.50 

164 

173 

.8 

0 

20 

98 

107 

116 

125 

134 

143 

: 152 

101 

170 

179 

•9 

7 

80 

102 

111 

120 

130 

139 

148 

i 157 

107 

176 

185 

81 

105 

115 

124 

134 

144 

153 

103 

172 

182 

191 

.1 

1 

32 

109 

119 

128 

138 

148 

1.58 

168 

178 

188 

198 

.2 

2 

33 

112 

122 

132 

143 

1.53 

163 

173 

183 

194 

204 

.3 

3 

84 

115 

126 

136 

147 

1.57 

168 

178 

189 

199 

210 

•4 

4 

35 

—119 

—130 

—140 

—151 

—102 

—173 

—184 

—194 

-205 

-216 

.5 

5 

30 

122 

133 

144 

156 

167 

178 

189 

200 

211 

222 

.6 

6 

37 

126 

137 

148 

100 

171 

183 

194 

200 

217 

228 

.7 

8 

38 

129 

141 

152 

164 

176 

188 

199 

211 

223 

235 

.8 

9 

30 

132 

144 

1.50 

109 

181 

193 

205 

217 

229 

241 

•9 

10 

40 

130 

148 

160 

173 

185 

198 

210 

222 

235 

247 

41 

139 

152 

165 

177 

190 

202 

215 

228 

240 

258 

. I 

1 

42 

143 

156 

169 

181 

194 

207 

220 

233 

246 

259 

.2 

3 

43 

146 

159 

173 

186 

199 

212 

226 

239 

252 

205 

.3 

4 

44 

149 

163 

177 

190 

204 

217 

231 

244 

258 

272 

•4 

6 

45 

—153 

-167 

-181 

—194 

-208 

-222 

-236 

—250 

—264 

—278 

.5 

7 

40 

156 

170 

185 

199 

213 

227 

241 

256 

270 

284 

.6 

8 

47 

160 

174 

189 

203 

218 

232 

247 

261 

276 

290 

.7 

10 

48 

163 

17-8 

193 

207 

222 

237 

252 

267 

281 

290 

.8 

11 

40 

160 

181 

197 

212 

227 

242 

257 

272 

287 

302 

•9 

13 

60 

170 

185 

201 

216 

231 

247 

262 

273 

293 

309 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

.1 

.2 

■3 

•4 

•5 

.6 

•7 

.8 

•9 

Corrections  for 

0 

1 

1 

0 

2 

3 

3 

4 

4 

tenths  in  width. 

TABLES. 


675 


TABLE  XI.  — Con  tin  ued. 

Volumes  by  the  Prismoidal  Formula. 


■5 

Heights. 

Corrections 
for  tenths 
in  height. 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

51 

173 

ISO 

205 

220 

236 

252 

268 

283 

299 

315 

. I 

2 

52 

177 

193 

209 

225 

241 

257 

273 

289 

305 

321 

.2 

3 

53 

180 

196 

213 

229 

245 

262 

278 

294 

311 

327 

.3 

5 

54 

183 

200 

217 

233 

250 

267 

283 

300 

317 

333 

•4 

7 

55 

—187 

-204 

—221 

—238 

—255 

-272 

—289 

—306 

—323 

-340 

.5 

8 

50 

190 

207 

225 

242 

259 

277 

294 

311 

328 

346 

.6 

10 

57 

104 

211 

229 

246 

264 

281 

299 

317 

334 

352 

.7 

12 

58 

197 

215 

233 

251 

269 

286 

304 

322 

340 

358 

.8 

14 

59 

200 

219 

237 

255 

273 

291 

310 

328 

346 

364 

•9 

15 

60 

204 

222 

241 

259 

278 

296 

315 

333 

352 

370 

61 

207 

226 

245 

264 

282 

301 

320 

339 

358 

377 

. I 

2 

62 

210 

230 

249 

268 

287 

306 

325 

344 

364 

383 

.2 

4 

63 

214 

233 

253 

272 

292 

311 

331 

350 

369 

389 

.3 

6 

64 

217 

237 

257 

277 

296 

316 

336 

356 

375 

395 

•4 

8 

65 

—221 

-241 

—261 

-281 

-301 

—321 

-341 

-361 

—381 

-401 

.5 

10 

66 

224 

244 

265 

285 

306 

326 

346 

367 

387 

407 

.6 

12 

67 

227 

248 

269 

290 

310 

331 

352 

37'2 

393 

414 

.7 

14 

68 

231 

252 

273 

294 

315 

336 

357 

378 

399 

420 

.8 

16 

69 

234 

256 

277 

298 

319 

311 

362 

383 

405 

426 

•9 

18 

70 

238 

259 

281 

302 

324 

346 

367 

389 

410 

432 

71 

241 

263 

285 

307 

329 

351 

373 

394 

416 

438 

. I 

2 

72 

244 

267 

289 

311 

333 

356 

378 

400 

422 

444 

.2 

5 

73 

248 

270 

293 

315 

338 

360 

383 

406 

428 

451 

.3 

7 

74 

251 

274 

297 

320 

313 

365 

388 

411 

434 

457 

■4 

9 

o_-55 

— 278 

—301 

—324 

—347 

—370 

—394 

417 

—440 

4(33 

12 

i 0 

76 

258 

281 

305 

328 

352 

375 

399 

422 

446 

469 

• 5 
.6 

14 

77 

2(il 

285 

309 

333 

356 

380 

404 

428 

452 

475 

.7 

16 

78 

2(55 

289 

• 313 

337 

361 

385 

409 

433 

457 

481 

.8 

19 

79 

2(58 

293 

317 

341 

366 

390 

415 

439 

463 

488 

•9 

21 

80 

272 

296 

321 

346 

370 

395 

420 

444 

469 

494 

81 

275 

300 

325 

350 

375 

400 

425 

450 

475 

500 

. I 

3 

82 

278 

304 

329 

354 

380 

405 

430 

456 

481 

506 

.2 

5 

83 

282 

307 

333 

359 

384 

410 

435 

461 

487 

512 

.3 

8 

84 

285 

311 

337 

363 

389 

415 

441 

467 

493 

519 

•4 

10 

85 

—289 

—315 

-341 

-367 

-394 

—420 

—446 

-472 

—498 

-525 

.5 

13 

86 

292 

319 

345 

372 

398 

425 

451 

478 

504 

531 

.6 

16 

87 

295 

322 

349 

376 

403 

430 

456 

483 

510 

537 

• 7 

18 

88 

299 

326 

353 

380 

407 

435 

462 

489 

516 

543 

.8 

21 

89 

303 

330 

357 

385 

412 

440 

467 

494 

522 

549 

•9 

24 

90 

306 

333 

361 

389 

417 

444 

472 

500 

528 

556 

91 

309 

337 

365 

393 

421 

449 

477 

506 

534 

562 

.1 

3 

92 

312 

311 

369 

398 

426 

454 

483 

511 

540 

568 

.2 

0 

93 

316 

314 

373 

402 

431 

459 

488 

517 

545 

574 

.3 

9 

94 

319 

348 

377 

406 

435 

464 

493 

522 

551 

580 

•4 

12 

95 

-323 

—352 

-381 

—410 

—440 

-469 

—498 

-528 

-557 

—586 

.5 

15. 

96 

326 

356 

385 

415 

444 

474 

504 

533 

563 

593 

.6 

18 

97 

329 

359 

389 

419 

449 

479 

509 

539 

569 

599 

.7 

21 

98 

333 

3(>3 

393 

423 

454 

484 

514 

514 

575 

605 

.8 

23 

99 

336 

367 

397 

428 

458 

489 

519 

550 

581 

Oil 

•9 

26, 

100 

340 

370 

401 

432 

463 

494 

525 

556 

586 

617 

11 

12 

13 

i 14  ' 

15 

16 

17 

18 

19 

. I 

.2 

l_^ 

1 -4 

•5 

iIL!| 

•7 

.8 

•9 

Corrections  for 

0 

1 

1 ‘ 

2 

2 

3 

3 

^ 1 

4 

tenths  in  width. 

676 


SUR  VE  YJNG. 


TABLE  XI. — Continued. 


Volumes  by  the  Prismoidal  Formula. 


Widths. 

1 1 RIGHTS. 

(Correct  ions 
for  tcnilis 

III  lici(;lit. 

21 

22 

23 

a, 

25  j 

26 

27 

28 

20 

30 

1 

6 

7 

7 

7 

8 

8 

9 1 

9 

9 

. I 

0 

2 

13 

14 

14 

15 

15 

10 

17 

17  j 

18 

19 

. a 

0 

8 

19 

20 

21 

22 

23 

21 

2.5 

20  1 

27 

28 

.3 

0 

4 

20 

27 

28 

30 

31 

32 

33 

35 

30 

37 

■4 

1 

5 

—32 

—34 

-ao 

-37 

—39 

—40 

—42 

—43 

—45 

—46 

.5 

1 

0 

39 

41 

43 

44 

40 

48 

50 

52 

54 

50 

.6 

1 

7 

45 

48 

50 

52 

54 

50 

58 

60 

63 

05 

.7 

8 

52 

54 

57 

59 

02 

04 

07 

09 

72 

74 

.8 

1 

9 

58 

01 

04 

07 

09 

72 

75 

78 

81 

83 

.9 

10 

05 

08 

71 

74 

77 

80 

83 

80 

90 

93 

11 

71 

75 

78 

81 

a5 

88 

92 

95 

98 

102 

. I 

0 

12 

78 

81 

85 

89 

93 

90 

100 

104 

107 

in 

.3 

1 

18 

84 

88 

92 

90 

100 

114 

108 

112 

no 

120 

.3 

1 

14 

91 

95 

99 

104 

108 

112 

117 

121 

125 

130 

■4 

2 

15 

-9? 

—102 

-100 

—111 

—no 

—120 

-125 

-130 

-1:44 

—139 

.5 

2 

10 

104 

109 

114 

119 

123 

128 

133 

1.38 

143 

148 

.6 

3 

17 

110 

115 

121 

120 

131 

130 

142 

147 

1.52 

1.57 

.7 

3 

18 

117 

122 

128 

133 

139 

144 

150 

1.50 

161 

107 

.8 

4 

19 

123 

129 

135 

111 

147 

1 52 

1.58 

104 

170 

170 

.9 

4 

20 

130 

130 

142 

148 

154 

100 

107 

173 

179 

185 

21 

130 

14? 

149 

150 

102 

109 

175 

181 

188 

194 

.1 

1 

22 

143 

149 

150 

103 

17’0 

177 

183 

190 

197 

204 

.2 

2 

28 

149 

150 

103 

170 

177 

185 

192 

199 

200 

213 

.3 

2 

24 

150 

103 

170 

178 

185 

193 

200 

207 

215 

222 

•4 

3 

25 

—102 

—170 

—177 

-185 

-193 

-201 

-208 

—210 

—224 

-231 

.5 

4 

20 

109 

177 

185 

193 

201 

209 

217 

225 

233 

241 

.6 

5 

27 

175 

183 

192 

200 

208 

217 

225 

233 

242 

2.50 

.7 

5 

28 

181 

190 

199 

207 

210 

225 

2733 

242 

251 

259 

.8 

6 

29 

188 

197 

200 

215 

224 

233 

242 

251 

260 

269 

•9 

7 

30 

194 

204 

213 

222 

231 

241 

250 

259 

209 

278 

31 

201 

210 

220 

230 

239 

249 

258 

208 

277 

287 

.1 

1 

32 

207 

217 

227 

237 

247 

257 

267 

277 

286 

290 

.2 

2 

38 

214 

224 

234 

214 

255 

205 

275 

285 

295 

31  lO 

.3 

3 

34 

220 

£31 

241 

252 

202 

273 

283 

294 

304 

315 

.4 

4 

35 

—227 

—238 

—248 

—259 

—270 

—281 

—292 

—302 

—313 

-324 

.5 

5 

30 

233 

244 

250 

207 

278 

289 

300 

311 

322 

333 

.6 

6 

37 

210 

251 

203 

274 

285 

297 

308 

320 

331 

.343 

.7 

8 

38 

240 

258 

270 

281 

293 

305 

317 

328 

340 

352 

.8 

9 

39 

253 

205 

277 

289 

3'1 

313 

325 

337 

349 

361 

•9 

10 

40 

259 

272 

284 

290 

309 

321 

333 

340 

358 

370 

41 

200 

278 

291 

304 

310 

329 

342 

354 

367 

380 

I 

1 

42 

272 

285 

298 

311 

324 

337 

350 

363 

376 

389 

.2 

3 

43 

279 

292 

305 

319 

332 

345 

358 

372 

385 

398 

.3 

4 

44 

285 

299 

312 

320 

340 

353 

307 

380 

. 394 

41 '7 

.4 

6 

45 

—292 

-300 

—319 

—333 

—347 

—301 

—375 

-389 

—403 

—417 

.5 

7 

40 

298 

312 

327 

341 

355 

309 

383 

398 

412 

420 

.6 

8 

47 

305 

319 

334 

348 

303 

377 

392 

406 

421 

435 

.7 

10 

48 

311 

320 

341 

350 

370 

385 

400 

415 

430 

444 

.8 

11 

49 

318 

333 

348 

303 

37'8 

393 

408 

423 

4.39 

4.54 

•9 

13 

50 

324 

340 

355 

370 

380 

401 

417 

432 

418 

463 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

. I 

.2 

•3 

•4 

•5 

.6 

•8 

•9 

Corrections  for 

1 

2 

2 

3 

4 

5 

5 

6 

7 

tenths  in  width. 

TABLES. 


677 


TABLE  XL — Continued, 


Volumes  by  the  Prismoidal  Formula. 


CO 

Heights. 

Corrections 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

111  height. 

61 

331 

346 

362 

378 

394 

409 

425 

441 

456 

472 

.1 

2 

52 

337 

353 

369 

385 

401 

417 

433 

449 

465 

481 

.2 

3 

53 

344 

360 

376 

393 

409 

425 

442 

458 

474 

491 

• 3 

5 

54 

350 

367 

383 

400 

417 

433 

450 

467 

483 

500 

•4 

7 

55 

—356 

—373 

-390 

—407 

—424 

—441 

-458 

—475 

—492 

—509 

• 5 

8 

56 

363 

380 

398 

415 

432 

449 

467 

484 

501 

519 

.6 

10 

57 

369 

387 

405 

422 

440 

457 

475 

493 

510 

528 

.7 

12 

68 

376 

394 

412 

430 

448 

465 

483 

501 

519 

537 

.8 

14 

59 

382 

401 

419 

437 

455 

413 

492 

510 

528 

546 

•9 

15 

60 

389 

407 

426 

444 

463 

481 

500 

519 

537 

556 

61 

395 

414 

433 

452 

471 

490 

508 

527 

546 

565 

.1 

2 

62 

402 

421 

440 

459 

478 

498 

517 

536 

555 

574 

.2 

4 

63 

403 

428 

447 

467 

486 

506 

525 

544 

564 

583 

.3 

6 

64 

415 

435 

454 

474 

494 

514 

533 

553 

573 

593 

•4 

8 

65 

—421 

—441 

—461 

—481 

—502 

-522 

—542 

-562 

—582 

—602 

.5 

10 

68 

428 

418 

469 

489 

509 

530 

550 

570 

591 

611 

.6 

12 

67 

431 

455 

476 

496 

517 

538 

558 

579 

600 

620 

• 7 

14 

68 

441 

462 

483 

504 

525 

546 

567 

588 

609 

630 

.8 

16 

69 

447 

469 

490 

511 

532 

554 

575 

596 

618 

639 

•9 

18 

70 

454 

475 

497 

519 

540 

562 

583 

605 

627 

648 

71 

460 

482 

504 

526 

548 

570 

592 

614 

635 

657 

.1 

2 

72 

467 

459 

511 

533 

556 

578 

600 

622 

644 

667 

.2 

5 

^ 0 

i 0 

473 

496 

518 

541 

563 

586 

608 

631 

653 

676 

.3 

7 

74 

430 

502 

525 

548 

571 

594 

617 

640 

662 

685 

.4 

9 

75 

—486 

—509 

—532 

-556 

—579 

-601 

—625 

-648 

—671 

-694 

• 5 

12 

76 

493 

516 

540 

563 

586 

610 

633 

657 

680 

704 

.6 

14 

77 

499 

523 

547 

570 

594 

618 

642 

665 

689 

713 

• 7 

16 

78 

506 

530 

554 

578 

602 

626 

650 

674 

698 

722 

.8 

19 

79 

512 

536 

561 

585 

610 

634 

658 

683 

707 

731 

•9 

21 

80 

519 

543 

568 

593 

617 

642 

667 

691 

716 

741 

81 

525 

550 

575 

600 

625 

650 

675 

700 

725 

750 

.1 

3 

82 

531 

557 

582 

607 

633 

658 

683 

709 

734 

759 

.2 

5 

83 

538 

564 

589 

615 

640 

666 

692 

717 

743 

769 

• 3 

8 

84 

544 

570 

596 

622 

648 

674 

700 

726 

752 

778 

•4 

10 

85 

— 551 

— 577 

-603 

-630 

-656 

—682 

—708 

—735 

—761 

-787 

• 5 

13 

86 

557 

584 

610 

637 

664 

690 

717 

743 

770 

796 

.6 

16 

87 

564 

591 

618 

644 

671 

698 

725 

752 

779 

806 

.7 

18 

88 

570 

598 

625 

652 

679 

706 

733 

760 

788 

815 

.8 

21 

89 

577 

004 

632 

659 

687 

714 

742 

769 

797 

824 

•9 

24 

90 

583 

611 

639 

667 

694 

722 

750 

777 

806 

833 

91 

590 

618 

646 

674 

702 

730 

758 

786 

815 

843 

. I 

3 

92 

596 

625 

653 

681 

710 

738 

767 

795 

823 

852 

.2 

6 

93 

603 

631 

060 

689 

718 

746 

775 

804 

832 

861 

• 3 

9 

94 

609 

638 

667 

696 

725 

754 

783 

812 

841 

870 

• 4 

12 

95 

—616 

-045 

-674 

—704 

-733 

— 762 

—792 

-821 

—850 

—880 

.5 

15 

96 

022 

652 

681 

711 

741 

770 

800 

830 

859 

889 

.6 

18 

97 

G29 

659 

689 

719 

748 

778 

808 

838 

868 

898 

• 7 

21 

00 

C35 

665 

696 

726 

756 

786 

817 

847 

877 

907 

.8 

23 

09 

C42 

672 

703 

733 

764 

794 

825 

856 

886 

917 

•9 

26 

ICO 

048 

679 

710 

741 

772 

802 

833 

864 

895 

926 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

, I 

.2 

■3 

•4 

•5 

.6 

•7 

.8 

•9 

Corrections  for 

1 

2 

2 

3 

4 

5 

5 

6 

7 

tenths  in  width. 

678 


SUR  VE  VI NG. 


TABLE  yA.  — Contifiucd. 


Volumes  by  the  Prismoidal  Formula. 


) 

Widths.  1 

Heights. 

Corrections 
for  tciulis 
in  heiKhl. 

31 

32 

33 

34 

35 

36 

37 

38 

39 

1 

1 

10 

10 

10 

10 

11 

11 

11 

12 

12 

12 

, 

0 

10 

20 

20 

21 

22 

22 

23 

23 

21 

25 

.2 

0 

ll 

2i) 

30 

31 

31 

32 

.33 

34 

35 

36 

; 37 

.3 

0 

4 

38 

40 

41 

42 

43 

44 

46 

47 

48 

49 

4 

1 

5 

—48 

—49 

— 51 

—52 

—54 

—56 

—57 

-.59 

—60 

1 —62 

,5 

1 

0 

57 

59 

61 

63 

65 

67 

68 

70 

72 

i "4 

.6 

1 

7 

67 

69 

71 

73 

76 

78 

80 

82 

81 

86 

.7 

1 

8 

't7 

79 

81 

84 

86 

89 

91 

94 

! 96 

, 97 

.8 

1 

9 

86 

89 

92 

94 

97 

100 

103 

106 

108 

1 111 

.9 

1 

10 

96 

99 

102 

105 

108 

111 

114 

117 

120 

: 123 

11 

10.5 

109 

112 

115 

119 

122 

126 

129 

1.32 

I 136 

. I 

0 

12 

115 

119 

122 

126 

1.30 

133 

137 

141 

1 14 

148 

.2 

1 

IS 

124 

128 

132 

136 

140 

144 

148 

1.52 

1.56 

■ 160 

.3 

1 

14 

134 

138 

143 

147 

151 

1.56 

160 

164 

169 

' 173 

.4 

2 

15 

—144 

—148 

—1.53 

—1.57 

—162 

—167 

—171 

—116 

— IHl 

1—185 

.5 

2 

16 

1.53 

1.58 

163 

168 

173 

178 

183 

188 

193 

198 

.6 

3 

17 

163 

168 

173 

178 

183 

189 

194 

199 

205 

210 

.7 

3 

18 

172 

178 

183 

189 

194 

200 

206 

211 

217 

222 

.8 

4 

19 

182 

188 

194 

199 

205 

211 

217 

223 

229 

235 

•9 

4 

20 

191 

198 

204 

210 

216 

222 

228 

235 

241 

247 

21 

201 

207 

214 

220 

227 

233 

240 

246 

253 

259 

. I 

1 

22 

210 

217 

224 

231 

238 

244 

251 

2.58 

265 

272 

.2 

2 

23 

220 

227 

234 

241 

248 

256 

263 

270 

277 

284 

.3 

2 

24 

230 

237 

244 

2.52 

259 

267 

274 

281 

289 

296 

•4 

.3 

25 

-239 

—247 

—25.5 

—262 

—270 

—278 

— 285 

—293 

—301 

—.3119 

.5 

4 

26 

249 

257 

265 

273 

281 

289 

297 

.305 

313 

321 

.6 

5 

27 

2.58 

267 

275 

283 

292 

300 

.308 

317 

325 

333 

.7 

5 

28 

268 

277 

285 

294 

302 

311 

320 

328 

.337 

346 

.8 

6 

29 

277 

286 

29.5 

304 

313 

322 

.331 

340 

349 

.358 

•9 

7 

30 

287 

296 

306 

315 

324 

333 

343 

352 

361 

370 

81 

297 

306 

316 

325 

335 

344 

354 

.364 

373 

383 

.1 

1 

32 

306 

316 

326 

336 

346 

356 

365 

375 

385 

395 

.2 

0 

33 

316 

326 

336 

346 

356 

367 

377 

387 

397 

407 

•3 

3 

34 

325 

336 

346 

357 

367 

378 

388 

.399 

409 

420 

•4 

4 

35 

-.335 

—346 

—356 

—367 

—378 

—389 

—400 

^10 

—421 

—432 

.5 

5 

36 

344 

356 

367 

378 

389 

400 

411 

422 

433 

444 

.6 

6 

37 

354 

365 

377 

388 

400 

411 

423 

434 

445 

4.57 

.7 

8 

38 

364 

375 

387 

399 

410 

422 

434 

446 

457 

469 

.8 

9 

39 

373 

.385 

397 

409 

421 

433 

445 

457 

469 

481 

•9 

10 

40 

383 

395 

407 

420 

4.32 

444 

457 

469 

481 

494 

41 

392 

405 

418 

4,30 

443 

456 

468 

481 

494 

.506 

j 

1 

42 

402 

415 

428 

441 

454 

467 

480 

493 

506 

519 

.2 

3 

43 

411 

425 

438 

451 

465 

478 

491 

504 

.518 

.531 

.3 

4 

44 

421 

4.35 

448 

462 

475 

489 

502 

516 

5.30 

.543 

■4 

6 

45 

—431 

—444 

—4.58 

-472 

—486 

—500 

—514 

—528 

— 542 

—556 

.5 

7 

46 

440 

4.54 

469 

483 

497 

511 

.525 

540 

.554 

568 

.6 

8 

47 

450 

464 

479 

493 

508 

522 

537 

.551 

566 

580 

.7 

10 

48 

4.59 

474 

489 

504 

519 

533 

548 

56.3 

578 

593 

.8 

11 

49 

469 

484 

499 

514 

.529 

.544 

560 

575 

590 

605 

.9 

13 

60 

478 

494 

.509 

525 

540 

556 

571 

586 

602 

617 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

. I 

.2 

•3 

•4 

•5 

.6 

•7 

.8 

•9 

Corrections  for 

1 

2 

3 

4 

5 

6 

8 

9 

10 

tenths  in  width. 

TABLES. 


679 


TABLE  XI. — Continued. 

Volumes  by  the  Prismoidal  Formula. 


CA 

-C 

Heights. 

Corrections 
for  tenths 

i 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

in  height. 

61 

488 

504 

519 

535 

551 

567 

582 

598 

614 

630 

. I 

2 

52 

498 

514 

530 

546 

562 

578 

594 

610 

626 

642 

.2 

3 

53 

507 

523 

540 

556 

573 

589 

005 

622 

638 

654 

.3 

5 

64 

517 

533 

550 

567 

583 

600 

617 

633 

650 

667 

•4 

7 

65 

— 526 

—543 

—560 

— 577 

—594 

—611 

—628 

—645 

-662 

—679 

.5 

8 

oC 

536 

553 

570 

588 

605 

622 

640 

657 

674 

691 

.6 

10 

67 

545 

563 

581 

598 

616 

633 

651 

669 

686 

704 

.7 

12 

58 

555 

573 

591 

609 

627 

644 

662 

680 

698 

716 

.8 

14 

69 

565 

583 

601 

619 

637 

656 

674 

692 

710 

728 

•9 

15 

60 

574 

593 

611 

630 

648 

667 

685 

704 

722 

741 

6] 

584 

602 

621 

640 

659 

678 

697 

715 

734 

753 

.1 

2 

62 

593 

612 

631 

651 

670 

089 

708 

727 

746 

765 

.2 

4 

63 

603 

622 

642 

661 

681 

700 

719 

739 

758 

778 

.3 

6 

64 

613 

632 

652 

672 

691 

711 

731 

751 

770 

790 

•4 

8 

65 

—623 

—642 

—662 

—682 

—702 

'J'OO 

—742 

—762 

—782 

—802 

.5 

10 

66 

631 

652 

672 

693 

713 

733 

754 

77’4 

794 

815 

.6 

12 

67 

641 

662 

682 

703 

724 

744 

765 

786 

806 

827 

.7 

14 

68 

651 

672 

693 

714 

735 

756 

777’ 

798 

819 

840 

.8 

16 

69 

660 

681 

703 

724 

745 

767 

788 

809 

831 

852 

•9 

18 

70 

670 

691 

713 

735 

756 

778 

799 

821 

843 

864 

71 

679 

701 

723 

745 

767 

789 

811 

833 

855 

877 

.1 

2 

72 

689 

711 

733 

756 

778 

800 

822 

844 

867 

889 

.2 

5 

73 

698 

721 

744 

766 

789 

811 

834 

856 

879 

901 

.3 

7 

74 

708 

731 

754 

777 

799 

822 

845 

868 

891 

914 

•4 

9 

76 

—718 

—741 

—764 

—787 

—810 

—833 

—856 

—880 

—903 

—926 

.5 

12 

76 

727 

751 

774 

798 

821 

844 

868 

891 

915 

938 

.6 

14 

77 

737 

760 

784 

808 

832 

856 

879 

903 

927 

951 

.7 

16 

78 

746 

710 

794 

819 

843 

867 

891 

915 

939 

963 

.8 

19 

79 

756 

780 

805 

829 

853 

878 

902 

927 

951 

975 

! -9 

21 

80 

765 

790 

815 

840 

864 

889 

914 

938 

963 

988 

81 

775 

800 

825 

850 

875 

900 

925 

950 

975 

1000 

.1 

3 

82 

785 

810 

835 

860 

886 

911 

936 

902 

987 

1012 

.2 

5 

83 

794 

820 

845 

871 

897 

922 

948 

973 

999 

1025 

.3 

8 

84 

804 

830 

856 

881 

907 

933 

959 

985 

1011 

1037 

•4 

10 

85 

—813 

—840 

—866 

—892 

—918 

—944 

—971 

—997 

—1023 

—1049 

.5 

13 

86 

823 

849 

876 

902 

929 

956 

982 

1009 

1035 

1062 

.6 

16 

87 

832 

859 

886 

913 

940 

967 

994 

1020 

1047 

1074 

.7 

18 

88' 

842 

809 

896 

923 

951 

978 

1005 

1032 

1059 

1086 

.8 

21 

89 

852 

87'9 

906 

. 934 

961 

989 

1016 

1044 

1071 

1098 

•9 

24 

90 

861 

889 

917 

944 

972 

1000 

1028 

1056 

1083 

nil 

91 

871 

899 

927 

955 

983 

1011 

1039 

1067 

1095 

1123 

.1 

3 

92 

880 

909 

937 

965 

994 

1022 

1051 

1079 

1107 

1136 

.2 

6 

93 

890 

919 

917 

976 

1005 

1033 

1062 

1091 

1119 

1148 

.3 

9 

94 

899 

928 

957 

986 

1015 

1044 

1073 

1102 

1131 

1160 

•4 

12 

95 

—909 

—938 

—968 

—997 

—1026 

-1056 

—1085 

-1114 

—1144 

—1173 

.5 

15 

96 

919 

948 

978 

1007 

1037 

1067 

1096 

1126 

1156 

1185 

.6 

18 

97 

928 

958 

988 

1018 

1048 

1078 

1108 

1138 

1168 

1198 

.7 

21 

98 

938 

968 

998 

1028 

1059 

1089 

1119 

1119 

1180 

1210 

.8 

23 

99 

947 

978 

1008 

1039 

1069 

1100 

1131 

1161 

1192 

1 222 

•9 

26 

100 

957 

988 

1019 

1049 

1080 

1111 

1142 

1173 

1204 

1235 

31 

32 

33 

34 

35 

36 

33 

39 

40 

. I 

.2 

•3 

•4 

•5 

.6 

•7 

.8 

■9 

Corrections  for 

1 

2 

3 

4 

5 

6 

8 

9 

10 

tenths  in  width. 

68o 


sun  VE  YtE  G. 


TABLE  XI. — Continued. 


Volumes  by  the  Prismoidal  Formula. 


Widths.  I 

1 

Heights, 

j Corrections 
j for  tenths 
' in  height. 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

1 

13 

13 

13 

14 

14 

14 

15 

15 

15 

15 

' J 

0 

2 

25 

20 

27 

27 

28 

28 

29 

80 

30 

31 

• 2 

0 

8 

38 

39 

40 

41 

42 

43 

44 

44 

45 

40 

; .3 

0 

4 

51 

52 

53 

54 

50 

57 

58 

59 

(K) 

02 

.4 

1 

6 

—03 

—05 

—00 

—08 

—09 

—71 

—73 

—74 

—76 

—77 

• 5 

1 

G 

70 

78 

80 

81 

83 

85 

87 

89 

91 

93 

.6 

1 

7 

89 

91 

93 

95 

97 

99 

102 

104 

106 

108 

.7 

1 

8 

101 

104 

106 

109 

111 

114 

110 

119 

121 

123 

; .8 

9 

111 

117 

119 

122 

125 

128 

131 

133 

136 

139 

i 

1 

10 

127 

130 

133 

136 

139 

142 

145 

148 

151 

154 

! 

11 

139 

113 

1 10 

149 

153 

156 

160 

103 

106 

170 

. I 

0 

12 

152 

156 

159 

103 

107 

170 

174 

178 

181 

185 

.2 

I 

18 

105 

109 

173 

177 

181 

185 

189 

193 

197 

201 

1 -3 

1 

14 

117 

181 

186 

190 

194 

199 

203 

207 

212 

216 

1 .4 

2 

l.> 

—190 

—194 

—199 

—204 

—208 

—213 

—218 

—222 

—227 

—231 

.5 

2 

IG 

203 

207 

212 

217 

222 

227 

232 

237 

242 

247 

.6 

3 

17 

215 

220 

226 

231 

236 

211 

247 

252 

257 

262 

.7 

3 

18 

228 

233 

239 

244 

250 

250 

201 

207 

272 

218 

.8 

4 

19 

210 

216 

252 

258 

204 

270 

270 

281 

2.S7 

293 

•9 

4 

20 

253 

259 

205 

272 

278 

284 

290 

290 

302 

309 

21 

266 

272 

279 

285 

292 

298 

305 

311 

318 

324 

. I 

1 

22 

278 

285 

292 

299 

306 

312 

319 

320 

333 

340 

.2 

2 

23 

291 

298 

305 

312 

319 

327 

334 

341 

318 

355 

.3 

2 

24 

304 

311 

319 

326 

333 

341 

348 

356 

363 

370 

•4 

3 

25 

—316 

-324 

—332 

— :340 

-347 

—355 

—363 

—370 

-378 

—386 

.5 

4 

2G 

3J9 

337 

315 

353 

361 

369 

377 

385 

393 

401 

.6 

5 

27 

342 

310 

358 

367 

375 

383 

392 

400 

408 

417 

.7 

5 

28 

354 

303 

372 

380 

389 

398 

406 

415 

423 

432 

.8 

6 

29 

367 

376 

385 

394 

403 

412 

421 

430 

439 

448 

•9 

7 

80 

380 

389 

393 

407 

417 

426 

435 

444 

454 

463 

31 

392 

402 

411 

421 

431 

410 

450 

459 

409 

478 

. I 

t 

82 

405 

415 

425 

435 

444 

454 

464 

474 

484 

494 

.2 

2 

33 

418 

428 

438 

448 

458 

469 

479 

489 

499 

509 

.3 

3 

84 

430 

441 

451 

402 

472 

483 

493 

504 

514 

525 

•4 

4 

35 

—443 

—454 

— 405 

— 175 

-486 

—497 

—508 

— 519 

—529 

—540 

.5 

5 

36 

456 

407 

478 

489 

500 

511 

522 

533 

544 

556 

.6 

6 

37 

408 

480 

491 

502 

514 

525 

537 

548 

560 

671 

.7 

. 8 

38 

481 

493 

504 

516 

528 

540 

551 

563 

575 

586 

.8 

9 

39 

494 

506 

518 

530 

542 

554 

506 

578 

590 

602 

• -9 

10 

40 

506 

519 

531 

543 

556 

568 

580 

593 

605 

617 

41 

519 

531 

544 

557 

569 

582 

595 

607 

620 

633 

. I 

1 

42 

531 

544 

557 

570 

583 

596 

609 

622 

635 

648 

.2 

3 

43 

544 

557 

571 

584 

597 

610 

624 

637 

650 

664 

.3 

4 

44 

557 

570 

584 

598 

Oil 

625 

638 

652 

665 

679 

•4 

6 

45 

—509 

—583 

—597 

—Oil 

-625 

—639 

— 653 

—667 

—681 

—094 

.5 

7 

4G 

582 

596 

010 

625 

639 

053 

607 

681 

696 

710 

.6 

8 

47 

595 

009 

624 

038 

653 

007 

082 

096 

711 

725 

.7 

10 

48 

007 

622 

037 

052 

667 

081 

096 

711 

726 

741 

.8 

11 

49 

020 

035 

050 

005 

681 

090 

710 

726 

741 

756 

•9 

13 

60 

633 

018 

004 

019 

694 

710 

725 

741 

756 

772 

41 

42 

44 

45 

46 

47 

48 

49 

50 

.1 

. 2 

■3 

•4 

■5 

.6 

■7 

.8 

•9 

Corrections  for 

1 

3 

4 

0 

7 

8 

10 

11 

13 

tenths  in  width. 

TABLES. 


68 1 


TABLE  XI. — Continued. 

Volumes  by  the  Prismoidal  Formula. 


cn 

Heights. 

Corrections 

41 

42 

43 

44 

45 

46 

47 

48 

49 

60 

in  height. 

61 

645 

661 

677 

693 

708 

724 

740 

756 

771 

787 

2 

52 

658 

674 

690 

706 

722 

738 

754 

770 

786 

802 

.2 

3 

53 

671 

687 

703 

720 

736 

752 

768 

785 

802 

818 

• 3 

5 

54 

683 

700 

717 

733 

750 

767 

783 

800 

817 

833 

*4 

7 

65 

—696 

—713 

—730 

—747 

—764 

—781 

—798 

—815 

-832 

—849 

• 5 

8 

66 

709 

726 

743 

760 

778 

795 

812 

830 

847 

864 

.6 

10 

57 

721 

739 

756 

774 

792 

809 

827 

844 

862 

880 

• 7 

12 

58 

734 

752 

770 

788 

806 

823 

841 

859 

877 

895 

.8 

14 

59 

747 

765 

783 

801 

819 

833 

856 

874 

892 

910 

•9 

15 

60 

759 

778 

796 

815 

833 

852 

870 

889 

907 

926 

61 

772 

791 

810 

828 

847 

866 

885 

991 

923 

941 

. I 

2 

62 

785 

804 

823 

842 

861 

880 

899 

919 

938 

957 

.2 

4 

63 

797 

817 

836 

856 

875 

894 

914 

933 

953 

972 

• 3 

6 

64 

810 

830 

849 

869 

889 

909 

928 

948 

968 

988 

•4 

8 

65 

—823 

-843 

—863 

—883 

—903 

—923 

-943 

—963 

—983 

—1003 

• 5 

10 

66 

835 

856 

876 

896 

917 

937 

957 

978 

998 

1019 

.6 

12 

67 

848 

869 

889 

910 

931 

951 

972 

993 

1013 

1034 

. 7 

14 

68 

860 

881 

902 

923 

944 

965 

986 

1007 

1028 

1049 

.8 

16 

69 

873 

894 

916 

937 

958 

980 

1001 

1022 

1044 

1065 

•9 

18 

70 

886 

907 

929 

951 

972 

994 

1015 

1037 

1059 

1080 

71 

898 

920 

942 

964 

986 

1008 

1030 

1052 

1074 

1096 

. I 

2 

72 

911 

933 

956 

978 

1000 

1022 

1044 

1067 

1089 

nil 

.2 

3 

73 

924 

946 

969 

991 

1014 

1036 

1059 

1081 

1104 

1127 

• 3 

7 

74 

936 

959 

982 

1005 

1028 

1051 

1073 

1096 

1119 

1142 

•4 

9 

75 

—949 

—972 

—995 

—1019 

—1042 

—1065 

-1088 

—nil 

-1134 

—1157 

• 5 

12 

76 

962 

985 

1009 

1032 

1056 

1079 

1102 

1126 

1149 

1113 

.6 

14 

77 

974 

998 

1022 

1046 

1069 

1093 

1117 

1141 

1165 

1188 

• 7 

16 

78 

987 

1011 

1035 

1059 

1083 

1107 

1131 

1156 

1180 

1204 

.8 

19 

79 

1000 

1024 

1048 

1073 

1097 

1122 

1146 

1170 

1195 

1219 

•9 

21 

80 

1012 

1037 

1062 

1086 

nil 

1136 

1160 

1185 

1210 

1235 

81 

1025 

1050 

1075 

1100 

1125 

1150 

1175 

1200 

1225 

1250 

. I 

3 

82 

1038 

1063 

1088 

1114 

1139 

1164 

1190 

1215 

1240 

1265 

.2 

5 

83 

1050 

1076 

1102 

1127 

1153 

1178 

1204 

1230 

1255 

1281 

• 3 

8 

84 

1063 

1089 

1115 

1141 

1167 

1193 

1219 

1244 

1270 

1296 

•4 

10 

85 

—1076 

—1102 

—1128 

—1154 

—1181 

—1207 

—1233 

—1259 

—1285 

—1312 

• 5 

13 

86  ' 

1088 

1115 

1141 

1168 

1194 

1221 

1248 

1274 

1301 

1327 

.6 

16 

87  1 

1101 

1128 

1155 

1181 

1208 

1235 

1262 

1289 

1316 

1343 

• 7 

18 

88 

1114 

1141 

1168 

1195 

1222 

1249 

1277 

1304 

1331 

1358 

.8 

21 

89 

1126 

1154 

1181 

1209 

1236 

1264 

1291 

1319 

1346 

1373 

•9 

24 

90 

1139 

1167 

1194 

1222 

1250 

1278 

1306 

1333 

1361 

1389 

91 

1152 

1180 

1208 

1236 

1264 

1292 

1320 

1348 

1376 

1404 

.1 

3 

92 

1164 

1193 

1221 

1249 

1278 

1306 

1335 

1363 

1391 

1420 

.2 

6 

93 

1177 

1206 

1234 

1263 

1292 

1320 

1349 

1378 

1406 

1435 

• 3 

9 

94 

1190 

1219 

1248 

1277 

1306 

1335 

1364 

1393 

1422 

1451 

• 4 

12 

95 

—1202 

—1231 

—1261 

-1290 

—1319 

—1349 

—1378 

-1407 

—1437 

—1466 

5 

15 

96 

1215 

1244 

1274 

1304 

1333 

1363 

1393 

1422 

1452 

1481 

.6 

18 

97 

1227 

1257 

1287 

1317 

1347 

1377 

1407 

1437 

1467 

1497 

• 7 

21 

98 

1240 

1270 

1301 

1331 

1361 

1391 

1422 

1452 

1482 

1512 

.8 

23 

99 

1253 

1283 

1314 

1344 

1375 

1406 

1436 

1467 

1497 

1528 

•9 

26 

100 

1265 

1296 

1327 

1358 

1389 

1420 

1451 

1481 

1512 

1543 

41 

42 

43 

44 

45 

46 

47 

48 

49 

60 

.1 

.2 

•3 

■4 

•5 

.6 

•7 

.8 

•9  1 

Corrections  for 

3 

4 

6 

7 

8 

10 

11 

13 

tenths  in  width. 

INDEX. 


Abney  Level  and  Clinometer 

Accuracy  of  the  Stadia  Method 

Attainable  by  Steel  Tapes,  and  Metallic  Wires  in  Measurements 

Adjustments,  Method  of  Studying 

General  Principle  of  Reversion 

of  Compass 

of  Level 

Precise  

of  Plane  Table 

of  Sextant 

of  Solar  Compass 

Attachment 

of  Transit 

of  Angles  in  Triangulation  Systems 

Triangle 

Quadrilateral..  . 

Larger  Systems 

of  Polygonal  Systems  in  Leveling 

Agreement,  Want  of,  between  .Surveyors 

Alignment,  Corrections  for,  in  Base-line  Measurements 

to  invisible  Stations 

Altitude  of  a Heavenly  Body 

Aneroid  Barometer 

Angle  Measurement  in  Triangulation 

Angles  Measured  by  Chain 

Angular  Measurements  in  Subdivision 

Areas  of  Cross  Sections  in  Rivers 


PAGE 

141 

263 

473 

4 

15 

15 

63 

553 

119 

Ill 

41 

102 

86 

491 

493 

494 

506 

559 

393 

462 

432 

531 

127 

477  to  488 

12 

360 

290 


684 


INDEX. 


PAGE 

Areas  of  Land 179 

by  Triangular  Subdivision  180 

from  boundary  Lines . . 181 

from  Rectangular  Coordinates  of  the  Corners 200 

of  Irregular  Figures 208 

Formulx  for  Derived 605 

Azimuth  Defined ii 

and  Latitude  by  Observations  on  Circumpolar  Stars 508  to  518 

of  Polaris  at  Elongation,  Table  of 33 

Balancing  a Survey 190 

Barometer,  Aneroid 127 

use  of  the  Aneroid 136 

Barometric  Formula  Derived 128 

Tables I33 

Base-line  and  its  Connections 427 

Measurement 447  to  465 

Broken,  Reduction  to  a Straight  Line ....  468 

Reduction  to  Sea  Level 468 

Computation  of  Unmeasured  Portion 472 

Summary  of  Corrections  to 469 

Bed  Ownership  in  Water  Fronts 586 

Bench-marks 74,  291 

in  Cities 384 

in  Triangulation 445 

Borrow  Pits 420 

Bubble,  Value  of  one  Division  of 58,  550 

Bubbles,  Level 55 

Construction  of  Tube 5^ 

Propositions  Concerning 57 

Use  of,  in  Measuring  Small  Vertical  Angles 5^ 

Angular  Value  of  one  Division  found  in  three  ways 5^ 

General  Considerations 59 

Buoys  and  Buoy  Flags 283 

Catenary  Effect  with  Steel  Tapes 457  to  465 

Chain,  Engineer’s  5 

Gunter’s 5 

Erroneous  Lengths  of - 6 

Testing  of 6 

Permanent  Provision  for 7 


INDEX. 


685 


PAGE 

Chain,  Standard  Temperature 7 

Use  of 8 

On  Level  Ground 8 

On  Uneven  Ground 8 

Number  and  use  of  Pins g 

Exercises  with ii 

Chaining  over  a Hill il 

Across  a Valley ii 

Random  Lines ii 

Check  Readings  in  Topographical  Surveying 253 

Circumpolar  Stars,  Times  of  Elongation  and  Culmination 510 

Pole  distances  of 511 

Azimuth  of  Polaris  at  Elongation 33 

City  Surveying 356 

Land  Surveying  Methods  Inadequate 356 

The  Transit 357 

The  Steel  Tape 357 

Laying  out  a Town  Site 359 

Provision  for  Growth 359 

Contour  Maps  36a,,  371 

Angular  Measurements  in  Subdivision 360 

I^aying  out  the  Ground 36  r 

Plat  to  be  Geometrically  Consistent 363 

Monuments 363 

Surveys  for  Subdivision 365 

Datum  Plane 369 

Location  of  Streets 369 

Sewer  Systems 370 

Water  Supply 370 

Methods  of  Measurement 371 

Retracing  Lines  371 

Erroneous  Standards 372 

True  Standards 373 

Use  of  Tape 374 

Normal  Tension 376 

Working  Tension 380 

Effect  of  Wind 381 

Effect  of  Slope 382 

Temperature  Correction 382 

Checks 383 


686 


INDEX. 


PACK 

City  Surveying — Miscellaneous  Problems 384 

Improvement  of  Streets 384 

Permanent  Bench-marks 384 

Value  of  an  Existing  Monument 385 

Significance  of  Possession 387,  582 

Disturbed  Corners  and  Inconsistent  Plats 388 

Surplus  and  Deficiency 389,  584 

Investigation  and  Interpretation  of  Deeds 391 

Office  Records 391 

Preservation  of  Lines 392 

Want  of  Agreement  between  Surveyors 393 

Clinometer 141 

Coefficient  of  Expansion  of  Steel  Tapes 456 

of  Brass  Wires 456 

Compass,  Needle,  Description  of 13 

Adjustments 15 

Use  of 34 

Setting  of  the  Declination 36 

Local  Attractions,  Sources  of 36 

Tests  of 37 

To  Establish  a Line  of  a Given  Bearing 37 

To  Find  a True  Bearing  of  a Line 37 

To  Retrace  an  Old  Line 37 

Exercises  with  Compass  and  Chain 38 

Compass,  Prismatic  Pocket 38 

Compass,  Solar 39 

Adjustments  of 41 

Use  of 44 

Finding  the  Declination  of  the  Sun 44 

Errors  in  Azimuth  due  to  Errors  in  Declination  and  Latitude 49 

Table  of  such  Errors 51 

Time  of  Day  Suitable  for  Observations 52 

Exercises  with  the  Solar  Compass 53 

Convergence  of  Meridians 176,  574 

Contour  Lines,  Propositions  Concerning 260 

Found  by  Transit  and  Stadia 259 

Clinometer 275 

Used  in  Computing  Earth  Work 379 

Contour  Maps  in  City  Work. ...  360,  371 

Coordinate  Protractor  168 

Corner  Monuments  in  Land  Surveying 178,  580 


INDEX. 


687 


PAGE 

Comer  Monuments  in  Land  Surveying — Cannot  be  Established  by  Surveyors.  581 

Cross-sectioning  in  Earth-work 406 

Cross-sections,  Areas  of,  in  Rivers 2go 

of  Least  Resistance 328 

in  Earth-work 404 

Cross-wires,  Illumination  of 518 

Setting  of 236 

Current-meters 300 

Rating  of 301 

Use  of,  in  Streams 300 

Conduits 313 

Curvature  and  Refraction,  Tables  of 433,  545 

Datum  Planes  in  Cities 369 

Declination  of  Magnetic  Needle 20 

Variations  in 20 

The  Daily  Variation 20 

The  Secular  Variation 21 

Other  Variations : 29 

To  Find  the  Declination  with  the  Compass  and  an  Observation  on  Po- 
laris   29 

Lines  of  equal  Declination  in  United  States,  or  Isogonic  Lines 23 

Formulae  for  finding  the  Declination  at  82  Points  in  the  United  States 

and  Canada 25 

Declination  of  the  Sun 44 

Method  of  Finding 44 

Correction  for  Refraction 45 

Table  of  Corrections 48 

Deeds,  Investigation  and  Interpretation  of 391 

Deficiency,  Treatment  of,  in  City  Work 389,  584 

Differences,  Finite,  Method  of 605 

Construction  of  Tables 605 

Derivation  of  Formulae  for  Evaluating  Irregular  Areas 608 

Direction  Meter 316 

Discharge  of  Streams 294 

Measuring  Mean  Velocities  of  Water  Currents 294 

Submerged  Floats 295 

Current  Meter 300 

Rating  the  Meter 301 

Rod  Floats 307 

Comparison  of  Methods 308 


688 


INDEX. 


PAGE 

Discharge  of  Streams — Relative  Rates  of  Flow  in  Different  Parts  of  the  Cross- 

section 309 

Computation  of  the  Mean  Velocity  over  the  Cross-section 312 

Sub-currents 316 

Flow  over  Weirs 316 

F'ormulae  and  Corrections 319 

Miner’s  Inch 322 

Flow  of  Water  in  Open  Channels,  Formula?  for 323 

Rutter’s  Formula?...  326 

Formula?  for  Brick  Conduits 327 

Cross-sections  of  Least  Resistance 328 

Sediment  Observations 329 

Collecting  the  Specimens 331 

Measuring  out  the  Samples 331 

Siphoning  off,  Filtering,  Weighing,  etc 332 

Disturbed  Corners 388 

Dredging 421 

Earth-work,  See  Volumes. 

Earth-work  Tables 410 

Elevation  of  Stations  in  Triangulation 432 

Elongation  of  Polaris,  Times  of ' 32 

EiTor,  Proportionate 2 

Errors,  Compensating  and  Cumulative 2 

in  Precise  Leveling 558 

Estimates,  Preliminary  in  Earth-work 399,  417 

Excavations  under  Water. 421 

Excess,  Spherical 494 

Expansion,  Coefficient  of 456 

Field  Notes,  Changes  in 3 

in  Land  Surveying 182 

in  Differential  Leveling 75 

in  Profile  Leveling 7^ 

in  Topographical  Surveying 249 

Filar  Micrometer 480 

Floats,  Submerged 295 

Flow  of  Water  in  Open  Channels 323 

in  Brick  Conduits 327 

Cross-sections  of  Least  Resistance 328 

See  also  Discharge  of  Streams. 


INDEX.  689 


PAGE 

Gauge  Hook 319 

Water 2gi,  563 

Geodetic  Leveling,  Trigonometrical  and  Spirit 540  to  563 

Geodetic  Positions,  Computation  of 535 

Derivation  of  Formulae  for 61 1 

Geodetic  Surveying 424 

Triangulation  Systems 425 

Base-line  and  its  Connections 427 

Reconnaissance 429 

Instrumental  Outfit  for 431 

Direction  of  Invisible  Stations 432 

Heights  of  Stations 432 

Construction  of  Stations 437 

Targets  ....  438 

Heliotropes 442 

Station  Marks 444 

Measurement  of  Base-lines 447 

Use  of  Steel  Tape  in 449 

Method  of  Mounting  and  Stretching 450 

M.  Jaderin’s  Method 453 

Absolute  Length 455 

Coefficient  of  Expansion 456 

Modulus  of  Elasticity 457 

Effect  of  Sag 457 

Temperature  Correction 459 

with  Metallic  Thermometer 460 

Correction  for  Alignment 462 

Sag 465 

Pull 465 

Reduction  of  Broken  Base  to  a Straight  Line 468 

Reduction  to  Sea  Level 468 

Summary  of  Corrections 469 

Computation  of  an  Unmeasured  Portion 472 

Accuracy  attainable  with  Steel  Tapes  and  Metallic  Wires 473 

Measurement  of  the  Angles 477 

Instruments 477 

Filar  Micrometer 480 

Programme  of  Observations 483 

Repealing  Method 484 

Continuous  Reading  around  the  Horizon 485 

Atmospheric  Conditions 487 

44 


690 


INDEX, 


PACK 

Geodetic  wSurveying — Geodetic  Night  Signals 488 

Reduction  to  the  Center 488 

Adjustment  of  the  Measured  Angles 491 

Equations  of  Condition 491 

Adjustment  of  a Triangle 493 

Spherical  Excess 494 

Adjustment  of  a Quadrilateral 494 

Geometrical  Conditions 494 

Angle  Equation  Adjustment 494 

Side  Equation  Adjustment 497 

Rigorous  Adjustment  for  Angle  and  Side  Equation 501 

Example  504 

Adjustment  of  Larger  Systems 506 

Computing  the  Sides  of  the  Triangles 506 

Latitude  and  Azimuth 508 

Conditions  of  the  Discussion 508 

Found  by  Observations  on  Circumpolar  Stars  at  Elongation  and  Cul- 
mination  508 

Observation  for  Latitude,  Two  Methods 512 

Correction  to  the  Meridian 514 

Observation  for  Azimuth 515 

Correction  to  Elongation 517 

The  Target » 518 

Illumination  of  Cross-wires 518 

Time  and  Longitude 519 

Fundamental  Relations 519 

Sidereal  to  Mean  Time 523 

Mean  to  Sidereal  Time 524 

Change  from  Sidereal  to  Mean  Time 525 

Observation  for  Time 526 

Selection  of  Stars 526 

List  of  Southern  Time  Stars 528 

Mean  Time  of  Transit 530 

Altitude  of  Star 531 

Making  the  Observations 532 

Programme  of  Observations 534 

Computing  the  Geodetic  Positions 535 

Table  of  L.  M.  Z.  Coefficients 537 

Example  539 

Geodetic  Leveling 540 

Trigonometrical  Leveling 54® 


INDEX. 


691 


PAGE 

Geodetic  Leveling — Formulae  for  Reciprocal  Observations 541 

Observations  at  one  Station  only 543 

an  observed  Angle  of  Depression 545 

Value  of  the  Coefficient  of  Refraction 546 

Precise  Spirit  Leveling 547 

Instruments 548 

Instrumental  Constants 550 

Daily  Adjustments 553 

Field  Methods 555 

Limits  of  Error 558 

Adjustment  of  Polygonal  Systems 559 

Determination  of  the  Elevation  of  Mean  Tide.. 563 

Grade,  Leveling  for 81 

Grading  over  Extended  Surfaces 396 

Hand  Level,  Locke’s 81 

Heights  of  Stations  in  Triangulation 432 

Heliotropes 442 

Hook  Gauge 319 

Horizontal  Angle  Measurement 93 

Hydrographic  Surveying  ....  277 

Location  of  Soundings  278 

Two  Angles  read  on  Shore 279 

in  the  Boat 279 

One  Range  and  one  Angle 282 

Buoys,  Buoy  Flags,  and  Range  Poles 283 

One  Range  and  Time  Intervals 284 

Intersecting  Ranges 284 

Cords  or  Wires 284 

Making  the  Soundings 285 

Lead.  285 

Line 285 

Sounding  Poles 287 

Soundings  in  Running  Water 287 

Water-surface  Plane  of  Reference 287 

Lines  of  Equal  Depth 288 

Soundings  on  Fixed  Cross-sections  in  Rivers 288 

Soundings  for  the  Study  of  Sand-waves 289 

Areas  of  Cross-section 290 

Bench-marks 291 

Water  Gauges 291 


692 


INDEX. 


PACK 

Hydrographic  Surveying — Water  Levels 292 

River  Slope 293 

Finding  the  Discharge  of  Streams  (See  Discharge  of  Streams) 294 

Illumination  OF  Cross-wires 518 

Inaccessible  Object. 

Distance  to  and  Elevation  oT 105 

Length  and  Bearing  of  a Line  joining  two  such 107 

Integrations  with  Current  Meter 301 

Isogonic  Lines  in  the  United  States 23  and  PI.  11. 

Judicial  Functions  of  Surveyors 579 

Kutter’s  Formul.,® 326 

Lakes,  Riparian  Rights  in 587 

Land  Surveying 172 

Laying  out  Land  172 

United  States  Method 173 

Origin  of 173 

Reference  Lines 173 

Division  into  Townships 174 

Division  into  Sections 175 

Convergence  of  Meridians 176 

Corner  Monuments 178 

Areas  of  Land 179 

by  Triangular  Subdivision 180 

by  use  of  Chain  alone 180 

by  use  of  Compass  or  Transit  and  Chain 180 

by  use  of  Transit  and  Stadia 181 

from  Bearing  and  Length  of  Boundary  Lines i8i 

Field  Notes 182 

Computing  the  Area 185 

The  Method  stated 185 

Latitudes,  Departures  and  Meridian  Distances 185 

Computing  Latitudes  and  Departures 187 

Balancing 190 

Rules  for  Balancing 192 

Error  of  Closure 193 

Form  of  Reduction. 194 

Area  Correction  Due  to  Erroneous  Length  of  Chain 197 


INDEX.  693 


PAGE 

Land  Surveying — From  Rectangular  Coordinates  of  Corners 200 

Conditions  of  Application 200 

M'ethod  Stated 201 

Form  of  Reduction 203 

Supplying  Missing  or  Erroneous  Data 203 

Bearing  and  Length  of  one  Course  unknown 205 

Bearing  of  one  Course  and  Length  of  Another  unknown 205 

Two  Bearings  unknown 206 

Lengths  of  two  Courses  unknown 206 

Plotting  the  Field  Notes 208 

Areas  of  Irregular  Figures 208 

Offsets  at  Irregular  Intervals 208 

Regular  Intervals 2io 

Subdivision  of  Land 213 

To  cut  off  by  a Line  through  a given  Point 213 

in  a given  Direction 215 

Exercises 220 

Latitude,  Geocentric  and  Geodetic 61 1 

Latitude  and  Azimuth 508 

Leads  used  in  Soundings 285 

Length,  Standards  of 372 

Absolute,  of  Steel  Tapes 455 

Lettering  on  Maps 575 

Level  Bubbles 55 

Level,  Hand 81 

Leveling,  Ordinary 7 1 

Precise  Spirit 547 

Trigonometric 540 

Leveling  Rods 70 

Levels,  Water 292 

Level  Surface ...  55 

Level,  The  Engineer’s 60 

Adjustments 63 

Relative  Importance  of 68 

Focussing  and  Parallax 68 

Use  of  the  Level 71 

Back  and  Fore-sights. 71 

Differential  Leveling 72 

Length  of  Sights 73 

Bench-marks 74 

The  Record 75 


694  INDEX. 


PACE 

Level,  DifTerential  Leveling — The  P'ield  Work 76 

Trofilc  Leveling 77 

The  Record 78 

Leveling  for  Grade 81 

Exercises 82 

Level  Trier  59 

Line,  Sounding 285 

Lines,  Clearing  out 432 

Preservation  of,  in  Cities 392 

L,  M.  Z.  Coefficients,  Table  of 537 

Formuloe  Derived 61 1 

Location  of  Railroad  on  Map 271 

Longitude,  Determination  of 519 

Map  Lettering 575 

Maps  in  Topographical  Survey 262 

in  Railroad  Surveying 267 

Projection  of 564 

Meander  Lines,  Extension  of,  in  Boundaries 584 

Mean  Tide,  Water,  Determination  of  Elevation  of,, 563 

Mean  Velocities  of  Water  Currents 294 

Measurement  of  Volumes 394 

Meridians,  Convergence  of 176,  574 

Metallic  Thermometer  Temperature  Corrections 460 

Micrometer,  Filar 480 

Mineral  Surveyors,  Instructions  to ■ 5^9 

Mining  Surveying 333 

Definitions 333 

Stations 335 

Instruments 335 

Mining  Claims 339 

Under-ground  Surveys 343 

To  Fix  the  Position  of  the  End  or  Breast  of  a Tunnel 343 

To  Find  the  Length  of  Tunnel  to  cut  a given  Vein 346 

To  Find  Direction  and  Distance  from  a Tunnel  to  a Shaft 348 

To  Survey  a Mine 35 1 

Missing  Data,  Supplying  of 203 

Bearing  and  Length  of  One  Course  unknown 205 

of  One  Course  and  Length  of  Another  unknown 205 

Two  Bearings  unknown 206 

Two  Lengths  unknown 206 


INDEX.  695 


PAGE 

Modulus  of  Elasticity  of  Steel  Tape 457 

Monuments  at  Section  Corners 178 

in  City  Work 363,  385,  388 

in  Triangulation 444 

Night  Signals  IN  Triangulatio.n 488 

Normal  Tension  of  Tape  in  City  Work 376 

Notes,  Field,  Changes  in 3 

Obstacles,  Passing  with  Chain  alone ii 

Transit  and  Chain 105 

Odometer 139 

Office  Records 391 

Offsets  in  Land  Surveying 208 

at  Regular  Intervals 208 

at  Irregular  Intervals 210 

Optical  Square 142 

Parallax,  How  Removed 68 

Parallel  Ruler 169 

Pantograph,  Theory  of 161 

Varieties 164 

Use  of 165 

Pedometer 137 

Pivot  Correction  in  Leveling 551 

Plane  Table 117 

Adjustments 119 

Use  of 120 

Location  by  Resection 123 

Resection  on  Three  Points 123 

Resection  on  Two  Points 124 

Use  of  Stadia 125 

Exercises 126 

Planimeters 143 

Theory  of  the  Polar  Planimeter 144 

To  Find  Length  of  Arm 150 

Suspended  Planimeter 152 

Rolling  Planimeter 152 

Theory  of 154 

Test  of  Accuracy  of  Planimeter  Measurements 157 

Use  of  the  Planimeter 158 


696 


INDEX. 


PACB 

rianimeters — Accuracy  of  Planimcter  Measurements 160 

Used  in  Computing  Earth-work 417 

Plats,  to  be  Geometrically  Consistent 363 

Inconsistent 388 

Platting  (See  Plotting). 

Plotting  in  Land  Surveying 208 

Topographical  Surveying 252,  254 

Railroad  Surveying 2O9 

Plumb-line,  its  great  Utility 55 

Use  of,  in  chaining 9 

Deviations  of 56 

Polaris,  Times  of  Elongation  of 32 

Azimuth  of,  at  Elongation 33 

Possession,  Significance  of 3S7,  582 

Preservation  of  Lines 392 

Prismoid,  The  Warped  Surface 408 

The  Henck’s. . 413 

Prismoidal  Forms 402 

Formulae 402 

Tables 410 

Precise  Spirit  Level 5<^7 

Projection  of  Maps 564 

Rectangular  Projection 564 

Trapezoidal  Projection 565 

Simple  Conic  Projection 566 

De  ITsle’s  Conic  Projection 567 

Bonne’s  Conic  Projection 567 

Poly  conic  Projection 568 

Formulae  used  in 568 

Derivation  of  Formulae 61 1 

Table  of  Constants. 571,  681 

Summary 572 

Convergence  of  Meridians 574 

Protractors 166 

Three-armed 167 

Paper  Protractor  167 

Coordinate  ....  168 

Topographical 255,  257 

Public  Lands  in  the  United  States 173 

(See  also  Land  Surveying). 


INDEX. 


697 


PAGE 

Railroad  Topographical  Surveying 265 

Objects  of  the  Survey 265 

P'ield  Work 265 

Another  Method 275 

Maps 267 

Plotting  the  Survey 269 

Making  the  Location  on  the  Map 271 

Ranges  and  Range  Poles  in  Sounding 284 

Reconnaissance  in  Triangulation 429  to  444 

Records,  Office,  in  City  Work 391 

P eduction  to  the  Center  in  Triangulation 488 

Refraction  and  Curvature,  Table  of  Values  of 433,  545 

Refraction,  Table  of  Mean  Values 512 

Tabular  Corrections  to  Declination  for,  with  Solar  Compass 47 

in  Trigonometrical  Leveling 540 

Coefficient  of 546 

Repeating,  Method  in  Triangulation 484 

Results,  Number  of  Significant  Figures  in 3 

Retracing  Lines  in  City  Work 371 

Riparian  Rights  in  Water  Fronts 584  to  587 

River  Slope 293 

Rod  Floats 307 

Ruler  Parallel 169 

Sag  Effect  with  Steel  Tapes 457,  465 

Sand  Waves,  Study  of 289 

Scales 169 

Sections  in  Land  Surveying 175 

Sediment  Observations 329 

Sewer  Systems * 370 

Sextant 108 

Theory no 

Adjustments in 

Use 112 

Exercises 112 

Shrinkage  of  Earth-work 420 

Sides,  Computation  of,  in  Triangulation 506 

Simpson’s  Rules  Derived 610 

Slope  of  River  Surface 293 

Solar  Attachments 99 

The  Saegmuller  Attachment 102 


698 


INDEX. 


PAf.E 

Solar  Attachments — Adjustments 102 

Solar  Compass  (See  Compass,  Solar). 

Soundings,  Location  of 278 

Making 285 

Spherical  Excess 494 

Stadia  Metliods,  Accuracy  of 263 

Stadia  Rod,  Craduation  of 237 

Stadia  Surveying  (See  Topographical  Surveying). 

Standards  of  Length  in  City  Work 372,  373 

Stars  for  Time  Determinations,  List  of 528 

Circumpolar,  Times  of  Elongation  and  Culmination  of 510 

Pole  Distances  of 511 

Stations,  Direction  of  Invisible 432 

Heights  of  in  Triangulation 432 

Construction  in  Triangulation 437 

Marks  at  in  Triangulation 444 

Steps,  Length  of  Men’s 138 

Steel  Tapes 9 

In  City  Work 357,  372  to  383 

In  Base-line  Measurement 449  to  465 

Straight  Lines  Run  by  Transit 95 

Streams,  Discharge  of 294 

Streets,  Location 369 

Improvement  of 384 

Stretch  of  Steel  Tapes 465 

Subdivision  of  Land 213 

Cutting  off  by  a Line  from  a given  Point 213 

in  a given  Direction 215 

Subdivision  of  Town  Plats 365 

Submerged  Floats 295 

Surplus,  Treatment  of,  in  City  Work 389,  584 

Surveying  Land  (See  Land  Surveying). 

Surveyors,  Want  of  Agreement - 393 

Judicial  Functions  of 579 

Cannot  Change  Original  Monuments 580 

The  Location  of  Lost  Monuments 580 

. Re-location  of  Extinct  Interior  Comers 580 

Cannot  “ Establish  ” Corners. 581 

Significance  of  Possession 582 

Surplus  and  Deficiency 584 

Meander  Lines,  Extension 584 


INDEX.  699 


PAGE 

Surveyors,  Meander  Lines,  not  Boundary  Lines 584 

Extension  of  Water  Fronts 585 

Bed  Ownership  in  Water  Fronts 586 

Riparian  Rights  in  Small  Lakes 587 

Tables,  Construction  of 605 

List  of 

I.  Trigonometric  Formulae 625 

II.  For  Converting  Meters,  Feet  and  Chains 629 

III.  Logarithms  of  Numbers  to  Four  Places 630 

IV.  Logarithmic  Traverse  Tables,  Four  Places 632 

V.  Stadia  Tables 640 

VI.  Natural  Sines  and  Cosines 648 

VII.  Tangents  and  Cotangents 656 

VIIL  Coordinates  in  Polyconic  Projections. 669 

IX.  Value  of  Coefficient  C in  Kutter’s  Formulae 670 

X.  Diameters  of  Brick  Conduits 671 

XL  Volumes  by  the  Prismoidal  Formulae 672 

Tape,  Steel  (See  Steel  Tape). 

Targets  in  Triangulation 438 

Temperature  Correction  in  Tapes 459 

Tension  of  Tape  in  City  Work 376,  380 

Tide  Water,  Determination  of  Elevation  of  Mean 563 

Time  and  Longitude,  Determination  of 519  to  534 

Time,  Sidereal  and  Mean 523 

Time  Stars,  List  of 528 

Three-point  Problem,  Four  Solutions 280 

Topographical  .Surveying 223 

Transit  and  Stadia  Method 224 

Fundamental  Relations 224 

Adaptation  to  Inclined  Sights 231 

Reduction  Tables  and  Diagrams 235 

Instrumental  Fixtures 236 

Setting  the  Cross  Wires 236 

Graduating  the  Stadia  Rod 237 

The  Topography 241 

Field  Work 241 

Reduction  of  Notes ..  249 

Plotting  the  Stadia  Line 252 

.Side  Readings 254 

Check  Readings. 253 


700 


INDEX. 


PAGB 

Topographical  Surveying — Contour  Lines 259 

The  I'inal  Map 262 

Topographical  Symbols 1C3 

Accuracy  of  the  Stadia  Method 263 

Topographical  vSymbols  263,  576 

Topography,  Railroad  (Sec  R.  R,  Topographical  Surveying). 

Townships  in  Land  Surveying 174 

Town  Site,  Laying  out 359 

Transit,  The  Engineer’s 83 

General  Description 83 

Adjustments 86 

Relative  Importance  of  Adjustments 89 

Eccentricity  in  Horizontal  Circle 90 

Inclination  of  Vertical  Axis 91 

Horizontal  Axis 92 

Collimation  Error 93 

Use  of  the  Transit 93 

Measurement  of  Horizontal  Angles 93 

Vertical  Angles 94 

Running  out  Straight  Lines 95 

Traversing 97 

The  Sola*-  Attachment 99 

Adjustments  of  Saegmuller  Attachment 102 

The  Gradienter  Attachment 104 

Care  of  the  Transit 104 

Exercises 105 

Transit  in  City  Work 357 

in  Mining  Work 336 

in  Topographical  Work 236 

Triangulation,  Instruments  used  in 477 

Programme  of  Observations 483 

Adjustment  of  Angles 491 

Computing  Sides  506 

Latitude  and  Azimuth 508 

Time  and  Longitude 5^9 

Computation  of  Geodetic  Positions 535 

Triangulation  Systems 425 

Traversing 97 

Trigonometer  (See  Coordinate  Protractor). 

Trigonometrical  Leveling 540 

Formulae 625 


INDEX.  701 


PAGE 

Variation  of  Magnetic  Needle  (See  Declination). 

Velocities  of  Water  Flow 294 

in  Vertical  Planes 301,  310 

in  Horizontal  Planes 309 

Verniers 18 

The  Smallest  Reading  of 20 

Rule  for  Reading 20 

Vertical  Angle  Measurement 94 

Volumes,  Measurement  of 394 

The  Elementary  Form 394 

Grading  over  Extended  Surfaces 396 

Approximate  Estimates  by  Means  of  Contours 399 

Prismoid 402 

Prismoidal  Formula 402 

Areas  of  Cross  Section 404 

Center  and  Side  Heights 405 

Area  of  Three-level  Sections 105 

Cross  Sectioning 406 

The  Warped  Surface  Prismoid 408 

Construction  of  Tables  for 410 

The  Henck  Prismoid 413 

Comparison  of  the  Henck  and  Warped  Surface  Prismoid 415 

Preliminary  Estimates  from  the  Profile 417 

Borrow  Pits 420 

Shrinkage  of  Earth-work 420 

Excavations  under  Water 421 

Water  Currents,  Mean  Velocity  of 294 

Sub-surface 316 

Water  Fronts,  Riparian  Rights  in 584 

Water  Gauges 291 

Water  Levels 292 

Water  Supply,  Surveys  for 370 

Weddel’s  Rule  Derived 610 

Weirs  Flow  Over 316 

Formulae  and  Corrections 319 

Wire  Interval,  Value  of 552 


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